In this letter, we derive the exact joint probability density function (pdf) of the amplitude and phase of the product of two correlated non-zero mean complex Gaussian random variables with arbitrary variances. This distribution is useful in many problems, for example radar and communication systems. We determine the joint pdf in terms of an infinite summation of modified Bessel functions of the first and second kinds, which generalizes the existing results. The truncation error is also studied when a truncated sum is employed. Finally, we evaluate the derived expressions through numerical experiments.