<UserControl x:Class="liblib.wpf.TextBoxDataList"
             xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation"
             xmlns:x="http://schemas.microsoft.com/winfx/2006/xaml"
             xmlns:mc="http://schemas.openxmlformats.org/markup-compatibility/2006"
             xmlns:d="http://schemas.microsoft.com/expression/blend/2008"
             xmlns:local="clr-namespace:liblib.wpf"
             xmlns:s="clr-namespace:System;assembly=mscorlib"
             mc:Ignorable="d">
    <Popup HorizontalOffset="0" VerticalOffset="1" PlacementTarget="{Binding RelativeSource={RelativeSource AncestorType=UserControl},Path=target}" IsOpen="{Binding RelativeSource={RelativeSource AncestorType=UserControl},Path=is_shown}" AllowsTransparency="True" StaysOpen="False">
        <Border BorderBrush="#2E84FF" BorderThickness="1" Background="#fff">
            <ListBox x:Name="select_listbox" ItemsSource="{Binding RelativeSource={RelativeSource AncestorType=UserControl},Path=available_data_items}" SelectionMode="Single" BorderThickness="0" HorizontalContentAlignment="Stretch" VerticalContentAlignment="Stretch" MaxHeight="150" ScrollViewer.VerticalScrollBarVisibility="Auto" ScrollViewer.CanContentScroll="False" AlternationCount="{x:Static s:Int32.MaxValue}">
                <ItemsControl.ItemTemplate>
                    <DataTemplate>
                        <Button Content="{Binding}" Click="item_Click" MouseEnter="item_MouseEnter" Focusable="False">
                            <Button.Template>
                                <ControlTemplate TargetType="Button">
                                    <TextBlock Text="{Binding}" Margin="8 1" />
                                </ControlTemplate>
                            </Button.Template>
                        </Button>
                    </DataTemplate>
                </ItemsControl.ItemTemplate>
            </ListBox>
        </Border>
    </Popup>
</UserControl>

using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;
using System.Threading;
using System.Threading.Tasks;
using System.Windows;
using System.Windows.Controls;
using System.Windows.Data;
using System.Windows.Documents;
using System.Windows.Input;
using System.Windows.Media;
using System.Windows.Media.Imaging;
using System.Windows.Navigation;
using System.Windows.Shapes;

namespace liblib.wpf
{
    /// <summary>
    /// TextBoxDataList.xaml の相互作用ロジック
    /// </summary>
    public partial class TextBoxDataList : UserControl
    {
        /// <summary>
        /// ターゲットの TextBox
        /// </summary>
        public TextBox target
        {
            get { return (TextBox)GetValue(targetProperty); }
            set { SetValue(targetProperty, value); }
        }

        // Using a DependencyProperty as the backing store for target.  This enables animation, styling, binding, etc...
        public static readonly DependencyProperty targetProperty =
            DependencyProperty.Register("target", typeof(TextBox), typeof(TextBoxDataList), new PropertyMetadata(null, (d, e) =>
            {
                var self = (TextBoxDataList)d;
                if (e.OldValue != null)
                {
                    // remove listeners
                    var old_textbox = (TextBox)e.OldValue;

                    old_textbox.LostFocus -= self.target_LostFocus;
                    old_textbox.PreviewKeyDown -= self.target_PreviewKeyDown;
                }

                // add listeners
                var new_textbox = (TextBox)e.NewValue;

                new_textbox.LostFocus += self.target_LostFocus;
                new_textbox.PreviewKeyDown += self.target_PreviewKeyDown;
            }));



        /// <summary>
        /// 表示するかどうか
        /// </summary>
        public bool is_shown
        {
            get { return (bool)GetValue(is_shownProperty); }
            set { SetValue(is_shownProperty, value); }
        }

        // Using a DependencyProperty as the backing store for is_shown.  This enables animation, styling, binding, etc...
        public static readonly DependencyProperty is_shownProperty =
            DependencyProperty.Register("is_shown", typeof(bool), typeof(TextBoxDataList), new PropertyMetadata(false));



        /// <summary>
        /// 全候補データ
        /// </summary>
        public IEnumerable<string> data_items
        {
            get { return (IEnumerable<string>)GetValue(data_itemsProperty); }
            set { SetValue(data_itemsProperty, value); }
        }

        // Using a DependencyProperty as the backing store for data_items.  This enables animation, styling, binding, etc...
        public static readonly DependencyProperty data_itemsProperty =
            DependencyProperty.Register("data_items", typeof(IEnumerable<string>), typeof(TextBoxDataList), new PropertyMetadata(null));



        /// <summary>
        /// 表示する候補データ
        /// </summary>
        public IEnumerable<string> available_data_items
        {
            get { return (IEnumerable<string>)GetValue(availble_data_itemsProperty); }
            set { SetValue(availble_data_itemsProperty, value); }
        }

        // Using a DependencyProperty as the backing store for availble_data_items.  This enables animation, styling, binding, etc...
        public static readonly DependencyProperty availble_data_itemsProperty =
            DependencyProperty.Register("available_data_items", typeof(IEnumerable<string>), typeof(TextBoxDataList), new PropertyMetadata(null));



        /// <summary>
        /// 自動で表示する項目を絞り込むか
        /// </summary>
        public bool auto_filtering
        {
            get { return (bool)GetValue(auto_filteringProperty); }
            set { SetValue(auto_filteringProperty, value); }
        }

        // Using a DependencyProperty as the backing store for auto_filtering.  This enables animation, styling, binding, etc...
        public static readonly DependencyProperty auto_filteringProperty =
            DependencyProperty.Register("auto_filtering", typeof(bool), typeof(TextBoxDataList), new PropertyMetadata(true));



        /// <summary>
        /// キータイプから操作更新までの遅延時間
        /// </summary>
        public int delay_msec
        {
            get { return (int)GetValue(delay_msecProperty); }
            set { SetValue(delay_msecProperty, value); }
        }

        // Using a DependencyProperty as the backing store for delay_msec.  This enables animation, styling, binding, etc...
        public static readonly DependencyProperty delay_msecProperty =
            DependencyProperty.Register("delay_msec", typeof(int), typeof(TextBoxDataList), new PropertyMetadata(500));



        /// <summary>
        /// 表示数
        /// </summary>
        public int display_number
        {
            get { return (int)GetValue(display_numberProperty); }
            set { SetValue(display_numberProperty, value); }
        }

        // Using a DependencyProperty as the backing store for display_number.  This enables animation, styling, binding, etc...
        public static readonly DependencyProperty display_numberProperty =
            DependencyProperty.Register("display_number", typeof(int), typeof(TextBoxDataList), new PropertyMetadata(5));



        private CancellationTokenSource token_source = null;

        /// <summary>
        /// コンストラクタ
        /// </summary>
        public TextBoxDataList()
        {
            InitializeComponent();
        }

        /// <summary>
        /// ターゲットでキー操作したときに候補を更新
        /// 絞り込みを更新してポップアップをオープン
        /// </summary>
        /// <param name="sender"></param>
        /// <param name="e"></param>
        private void target_PreviewKeyDown(object sender, KeyEventArgs e)
        {
            var items_num = this.available_data_items?.Count() ?? 0;

            switch (e.Key)
            {
                case Key.Escape:
                    this.is_shown = false;
                    break;

                case Key.Up:
                    if (items_num > 0)
                    {
                        this.select_listbox.SelectedIndex = (this.select_listbox.SelectedIndex + items_num - 1) % items_num;
                        this.select_listbox.ScrollIntoView(this.select_listbox.SelectedItem);
                    }
                    break;

                case Key.Down:
                    if (items_num > 0)
                    {
                        this.select_listbox.SelectedIndex = (this.select_listbox.SelectedIndex + 1) % items_num;
                        this.select_listbox.ScrollIntoView(this.select_listbox.SelectedItem);
                    }
                    break;

                case Key.Tab:
                case Key.Enter:
                    if (0 <= this.select_listbox.SelectedIndex && this.select_listbox.SelectedIndex < items_num)
                    {
                        this.complement((string)this.select_listbox.SelectedItem);
                    }
                    break;

                default:
                    this.deferredUpdate();
                    return;
            }

            e.Handled = true;
        }

        /// <summary>
        /// フォーカスはずれたときに閉じる
        /// </summary>
        /// <param name="sender"></param>
        /// <param name="e"></param>
        private void target_LostFocus(object sender, RoutedEventArgs e)
        {
            this.is_shown = false;
        }

        /// <summary>
        /// クリック
        /// </summary>
        /// <param name="sender"></param>
        /// <param name="e"></param>
        private void item_Click(object sender, RoutedEventArgs e)
        {
            var button = (Button)sender;
            this.complement((string)button.Content);
        }

        private void item_MouseEnter(object sender, MouseEventArgs e)
        {
            var button = (Button)sender;
            this.select_listbox.SelectedItem = button.Content;
        }

        /// <summary>
        /// テキストボックス更新
        /// </summary>
        /// <param name="text"></param>
        private void complement(string text)
        {
            this.target.Text = text;
            this.target.CaretIndex = text.Length;
            this.is_shown = false;
        }

        /// <summary>
        /// 遅延アップデート
        /// </summary>
        private void deferredUpdate()
        {
            // cancel previous scheduled updating
            this.token_source?.Cancel();

            this.token_source = new CancellationTokenSource();
            var token = token_source.Token;

            var delay_msec = this.delay_msec;

            // update with a little delay
            Task.Run(() =>
            {
                Thread.Sleep(delay_msec);
                token.ThrowIfCancellationRequested();
                this.Dispatcher.Invoke(() =>
                {
                    var text = this.target.Text;

                    this.available_data_items = this.data_items?.Where(x => x.Contains(text)).Distinct().Take(this.display_number).ToList();

                    this.is_shown = (this.available_data_items?.Count() ?? 0) > 0;
                    this.select_listbox.SelectedIndex = 0;
                });
            }, token);
        }
    }
}

<StackPanel Margin="10">
    <TextBox x:Name="textbox" Text="{Binding text}"/>
    <liblib:TextBoxDataList target="{Binding ElementName=textbox}" data_items="{Binding items}" display_number="10"/>
</StackPanel>

private class BindingData
{
    public List<string> items { get; set; }
        = new List<string> {
            "aa", "bbb", "acb", "abac",
            "abc", "cabac", "ddaeb", "cab",
            "bab", "abab", "abcab", "ab",
        };
}

public MainWindow()
{
    this.DataContext = new BindingData();
}