How does one define area on the complex plane?

Question

I have been thinking, that due to how functions and numbers can go onto the complex plane, and circles can be inscribed over that, such as that shown by e^{i * a}, and it leaves me wondering how exactly one would define and show area in this plane.

For example, what is the area of a unit circle in the complex plane.

Answer 1

Example

Using Green's theorem to find the area bounded by a closed contour $C$,

\begin{align} A &= \frac{1}{2} \oint_C x\, dy-y\, dx \\ &= \frac{1}{4i} \oint_C \bar z \, dz-z \, d\bar z \\ &= \frac{1}{2i} \oint_C \bar z \, dz \\ \end{align}