Does anyone know the name of the Win32 API that brings up this char picker when using the language bar?

Question

Answer 1

So the first answer attempt was a demonstration of missing the point.

To use the bar, Microsoft currently recommends using the Text Services Framework. When properly setup, your application can receive input from the IME without knowing or caring what IME is in use.

However, you can also use the Input Method Manager if you really want to dig into the API.