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 the 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.