Android Bottom Navigation Bar with drop shadow

You can draw your own shadow just above the bottom bar using simple View and its background:

<View
    android:layout_width="match_parent"
    android:layout_height="4dp"
    android:layout_above="@id/bottom_bar"
    android:background="@drawable/shadow"/>

drawable/shadow.xml:

<shape xmlns:android="http://schemas.android.com/apk/res/android">
    <gradient
        android:startColor="#1F000000"
        android:endColor="@android:color/transparent"
        android:angle="90" />
</shape>

Also, there are no compatibility issues if use this approach.

You can use elevation to add shadows to any view

<TextView
android:id="@+id/myview"
...
android:elevation="2dp"

android:background="@drawable/myrect" />

Refer this for more information

For those using a CoordinatorLayout with the Bottom Navigation Bar (or BottomAppBar), you can use the following to attach a shadow above the bar:

<View
    android:layout_width="match_parent"
    android:layout_height="4dp"
    android:background="@drawable/shadow"
    app:layout_anchor="@+id/toolbar"
    app:layout_anchorGravity="top"/>

Obviously, replace the @+id/toolbar with the id of the Bottom Navigation Bar