Is it possible to have a PhD advisor from another university? I am a graduate student, 2nd year, in a STEM field at a US university. I have passed my quals.  

Is it possible to ask from a Professor who is not at your university to be your advisor?  Did anybody hear of a similar situation?

I am very interested in the research of a professor who is at a nearby university, I really want to work with him but I don't know if this is even possible. (Maybe I should also note that his university is not necessarily better than the one I am now.) 

Any suggestions?
Extrapolating from the very few cases that I've seen, I think you'll need an official adviser at the school where you're actually enrolled.  If it's OK with that adviser, then you could also have a "real" adviser elsewhere, on an unofficial basis.  Your school may have some paperwork to be completed if your real adviser is to serve as a member of your official thesis committee.