I was reading a physics paper where it was mentioned that the basic framework of Connes' differential non-commutative geometry (or actually, a slight modification of Connes in that paper) would need some extensions in order to arrive at a theory of non-commutative complex geometry, where one now works with complex manifolds.

Are there specific technical obstacles which would make such an extension of non-commutative geometry particularly difficult or involved? What would be the main issues with trying to do this?