Let $X$ be a complex normal random variable. (Or, equivalently, a 2D real normal.) Is it possible to say anything useful about the distribution of the phase of $X$? Is it possible to do estimation on it?

What about the multivariate case? That is, I have a multivariate complex normal, and would like to understand the multivariate distribution of the phases of the individual components.

Does anything become easier if the real and imaginary parts are uncorrelated? (In the multivariate case, uncorrelated for each component.)