Consider $\mathbb{C}$ and let $dz=dx+idy$, $d\overline{z}=dx-idy$ be differential forms. If $\gamma$ is some path in $\mathbb{C}$, then one can interpret
$\int_\gamma f(z) dz$ as $\int_{0}^{1} f(\gamma(k)) dk$.
My question is how to interpret integrals over regions of the complex numbers. Specifically, I have seen the following construction for integrals over a region: Let $S\subset \mathbb{C}$ and write
$\int_S f(z) dz\wedge d\overline{z}$.
I can black box this to a certain extent, but I'd like to understand what is going on.
Question: How should this be interpreted? Why is the wedge product/conjugate present? Is there a more general construction?
You should interpret $S \subset \mathbb{C}$ as $S \subset \mathbb{R}^2$.
A complex valued differential form $\omega$ is by definition a real valued form plus $i$ times another real valued form. So $\omega = \omega_1 + i\omega_2$. The operations $d, \wedge, \int, $ etc. on real valued forms are extended to complex valued forms by linearity. Importantly, Stokes theorem holds for complex valued forms. As an example, you have seen the complex valued function $z : \mathbb{R}^2 \to \mathbb{C}$ defined by $z(x, y) = x + iy$. Hence $dz = dx + idy$. We can use the same extension procedure to define $\mathbb{R}^n$ valued differential forms.