The complex parabolic type Monge-Ampere equation we are dealing with is of the form (partial derivative u/partial derivative t) det[partial derivative(2)u/partial derivative z(i) ] = f in B x (0, T), u = phi on Gamma, where B is the unit ball in C-d, d > 1, and Gamma is the parabolic boundary of B x (0, T). Solution u is proved unique in the class C ((B) over bar x [0, T]) boolean AND W-infinity,loc(2.1) (B x (0, T)).