  We prove a bound for the Wasserstein distance between vectors of smooth
complex random variables and complex Gaussians in the framework of complex
Markov diffusion generators. For the special case of chaotic eigenfunctions,
this bound can be expressed in terms of certain fourth moments of the vector,
yielding a quantitative Fourth Moment Theorem for complex Gaussian
approximation on complex Markov diffusion chaos. This extends results of
Azmoodeh, Campese, Poly (2014) and Campese, Nourdin, Peccati (2015) for the
real case. Our main ingredients are a complex version of the so called
$\Gamma$-calculus and Stein's method for the multivariate complex Gaussian
distribution.
