Suppose $B$ is a bounded region in complex plane. In complex plane, one usually deals with complex moments, i.e. $\int_B {z}dxdy$ where $z \in \mathbb{C}$. What is so special about this complex moment compared to the real moments, i.e $\int_B {x^my^n}dxdy$ where $m,n \in \mathbb{N}$? I don see the reason using complex moments instead of real moments.

I'm not totally sure about what kind of answers you're looking for, but let me try something. First, let me emphasize that for a general Borel measure $\mu$ on the complex plane, the knowledge of the sequence $\int z^kd\mu(z)$, $k\in\mathbb N$ is not enough to characterize the measure $\mu$; you would instead need the real moments, or equivalently $\int z^k \bar z^\ell d\mu(z)$ for $k,\ell\in\mathbb N$. So, from a certain point of view, to work with the real moments you mentioned is actually important. On an other hand, as soon as you deal with orthogonal polynomials on the complex plane (e.g. orthogonal with respect to the area measure of some set $B$ as you consider), to deal with the complex moments is enough and more convenient: think about the GrammSchmidt procedure to get the coefficients of your orthogonal polynomials. So both this type of moments are of important use, depending on the context you're working with. 


For instance, in the setting of the logarithmic potential theory, what you "measure" are complex moments, aka harmonic moments. You can find more references e.g. here. 

