In this letter, we derive new results for the statistics of the ratio of two complex Gaussian random variables (RVs), where the numerator and denominator may have arbitrary means and are possibly correlated. Exact expressions are derived for the joint probability density function (pdf) of the real and imaginary parts, for the joint pdf of the amplitude and phase, and also for the joint characteristic function (cf) of the real and imaginary parts, which generalize the existing results. Then, we show an example application of the derived pdf to the symbol error probability (SEP) analysis for a single antenna communication system with imperfect channel state information (ICSI).