I would like if is possible to calculate the weak derivative of the complex absolute value function, i.e., $ f: \mathbb{C}\rightarrow \mathbb{R}^{+},$ where $f(z) = |z|^{2}, z\in \mathbb{C}$, by solving $$\int_{\mathbb{C}} fg' d\mu =-\int_{\mathbb{C}} vg d\mu, $$ where $v$ is the weak derivative of $f$ and the function $g$ is infinitely differentiable with compact support, i.e., $g\in C_{0}^{\infty}(\mathbb{C})$.
Thanks in advance.
Since you really do mean the complex derivative, compact supported smooth(i.e. entire) functions $g∈ C^∞_0(\mathbb C)$ are constant; compact support forces this constant to be 0 so any $v$ will do. When playing with complex derivatives we don't have bump functions or anything that comes with it, e.g. weak derivatives or partitions of unity.