yes it is. It is also related to the MGF of random variable. However, I don't know the MGF of the product of two complex gaussian random variable like the example above. Can we assume it as a chi-square or not?
moment generating function and cumulant generating function are very closely related ; i do not understand why you need the CGF for a product of random variables : terms like E { [X.X - E (X.X) ]2 } occur very naturally in the CGF or MGF of the random variable X (not X.X !)
As Fabrice said write the mgf and expand it or perhaps its log as a series. Since the pdf is a normal there will be a log somewhere and we know from calculus how to expand that. Then equate coefficients. This a standard method in Math. Stat.
See M.H, DeGroot, Probability and Statistics PP.162 ff.The mgf gets it name from the fact that the coeffs of its series expansion are the moments of the distribution.. So, mgf(t)=E(exp(tX)). Let A(t)=mgf(t). Then the nth derivative of A(t) [evaluated at t=0]= E(X^n) . Calculating an mgf is example 2 on p. 163-164. This is essentially your problem except your pdf is a bit more complicated. see pp. 217 ff. Best wishes, David