What Windows API should I use to create a blurred window?
See the question and my original answer on StackOverflowLayered Windows are an old thing. Technically on modern Windows, transparency like that is indeed achieved using the "Visual layer", aka "Direct Composition", the closest thing to the Desktop Windows Manager (the OS compositor).
This is a Windows' WinRT API so you can use it from any language with WinRT binding (Microsoft provides .NET and C++). This API is located in the Windows.UI.Composition WinRT namespace ("WUC").
The WinAppSDK with WinUI3 provides another implementation of a Direct Composition client in the Microsoft.UI.Composition ("MUC") namespace partially decoupled from Windows, this is today what Microsoft promote for what you want to do (again usable in .NET or C++).
This sample here https://github.com/aelyo-softworks/Wice demonstrates using WUC w/o any framework (it's C# but only using Win32/WinRT interop over WUC)
Note another solution for what you want to do is to host a WebView2 (you can host it using Direct Composition instead of the traditional HWND way) with transparency enabled and let the WebView use its CSS (this project does that again using C# but using 100% native API). This other sample demonstrates that https://github.com/smourier/ShellBat ("Acrylic Almond theme shows a 100% transparent WebView2), cf CoreWebView2Controller.DefaultBackgroundColor Property