  The purpose of this article is to determine explicitly the complete surfaces
with parallel mean curvature vector, both in the complex projective plane and
the complex hyperbolic plane. The main results are as follows: When the
curvature of the ambient space is positive, there exists a unique such surface
up to rigid motions of the target space. On the other hand, when the curvature
of the ambient space is negative, there are `non-trivial' complete parallel
mean curvature surfaces generated by Jacobi elliptic functions and they exhaust
such surfaces.
