The motivation for this question is a far-fetched dream I have where a lot of the excellent software available on *nix platforms could be trivially ported to Windows. Microsoft
Windows has already been such. The NT kernel itself has supported the concept of "personalities" (API layers over the NT layer) since the beginning, to support by design at the very least the Win32 API, the POSIX API and the OS/2 API.
The POSIX layer circulated for a long time in higher end SKUs (typically server-related) with different names (Microsoft POSIX subsystem/SFU/SUA), but it never really caught on for non-specialized use, both because it was not universally available (Microsoft never really pushed it, probably for commercial reasons) and because other solutions became widespread (think Cygwin/MSYS/MinGW).
Notice that, although the "personality API" is an interesting concept (and probably one of the cleanest way to implement a multi-API OS) it suffers a bit from a "deep segregation" problem - i.e., it's true that you can access kernel objects through a POSIX interface, but all services that have been built over the Win32 interface (like Windows, GDI & co.) aren't easily available; besides, however good the interface may be, there are some details (like the format of paths) that cannot be ironed over, so a POSIX application will always look a bit out of place.