Calculating a complex derivative of a polynomial What are the rules for derivatives with respect to $z$ and $\bar{z}$ in polynomials?
For instance, is it justified to calculate the partial derivatives of $f(z,\bar{z})=z^3-2z+\bar{z}-(\overline{z-3i})^4$ as if $z$ and $\bar{z}$ were independent? i.e. $f_z=3z^2-2$ and $f_\bar{z}=1-4(\overline{z-3i})^3$

no, for instance $f(z) = \overline z$ is not differentiable at any $z_0 \in \mathbb{C}$
