When someone speaks of integrating a complex function, it is always understood that they mean integrating along a path. However, there is another type of integration that is possible--namely integrating with the measure $dx \:dy$ over some area in the plane. Question: What applications are there of this latter type of integration on $\mathbb{C}$?
[I seem to recall there being some example of Yoccoz which used this kind of integration to solve a small denominator problem.]
One possible way to integrate on the complex plane is by considering complex measure. This is fairly similar to the method of Lebesgue integration on $\mathbb{R}^n$. For instance, we can simplify the integration by separating the real and complex portions. Allow $\mu_1$ and $\mu_2$ to be the real and complex parts of a complex measure (measurable function), with finite signed measure (can be negative). By the Hahn Decomposition Theorem, one can state $\mu_1=\mu_1^+-\mu_1^-$ and $\mu_2=\mu_2^+-\mu_2^-$.
Therefore, $\int_X{fd\mu}=(\int_X{fd\mu_1^+}-\int_X{fd\mu_1^-})+i(\int_X{fd\mu_2^+}-\int_X{fd\mu_2^-})\\=\int_X\mathfrak{R}(f)d\mu-i\int_X{\mathfrak{I}(f)d\mu}$
As you can see, we add the real and complex integrals, to obtain a complex measure for volume.