Let $M$ be a 4-manifold with a complex structure.

Does there exist a finite list of simply connected complex 4-manifolds $M_1, ... , M_n$ such that M is the quotient of some $M_i$ by the action of a group acting discretely on $M$?

This would be an analog of the Poincare-Koebe uniformization theorem in (real) dimension 2. People who I've asked this question to think that it's unlikely that there is such a list, but haven't been able to offer an argument or a reference.