N. R. Goodman  has discussed some aspects of the complex multivariate normal distribution, in particular, the analogue of the Wishart distribution and of multiple and partial correlations. We shall obtain maximum likelihood estimates for certain parameters and likelihood ratio tests of certain hypotheses arising in the study of such complex multivariate normal distributions. Although the general principles involved in the derivation of the distribution of the associated statistics are well known to mathematicians working on group representations (see, for example, Gelfand and Naimark , p. 24), it has seemed desirable to derive the needed results on this type in a way parallel to one method of obtaining the distributions of real multivariate analysis (Deemer and Olkin , Olkin  and Roy ). With the help of the results derived, we can handle the complex variates in the same manner as we do for real variates in the case of Gaussian distributions. Moreover, it can be noted that for every distributional result of classical multivariate Gaussian statistical analysis obtainable in closed (explicit) form, the counterpart analysis for complex Gaussian is also obtainable in closed (explicit) form with necessary changes. It may be pointed out that the non-central distributions in this connection were derived independently by A. T. James  with the help of zonal polynomials of hermitian matrices, but we feel that sometimes the derivation of distributions with the help of Jacobian transformations may be useful.
"Classical Statistical Analysis Based on a Certain Multivariate Complex Gaussian Distribution." Ann. Math. Statist. 36 (1) 98 - 114, February, 1965. https://doi.org/10.1214/aoms/1177700274