We study the generalized directed polymer model in which the amplitudes along a polymer assume random complex values. We introduce a replica trick which allows us to obtain fluctuations around mean values. It maps our model onto a system of n pairs of charged bosons. This system was first studied by Dyson, whose heuristic scaling theory provides prediction for the exponent of the transverse fluctuations of polymers with long-range correlated complex disorder. We show that complex disorder causes stronger transverse fluctuations than real positive disorder previously studied.