Which of the following is correct (considering $z = a+ib$):
1) Considering that $z$ and $z^*$ are independent $\frac{\partial |z|^2}{\partial z} = \frac{\partial (zz^*)}{\partial z} = z^*$
2) $\frac{\partial |z|^2}{\partial z} = \frac{\partial (a^2+b^2)}{\partial (a+ib)} = \frac{\partial (a^2+b^2)}{\partial a}\frac{\partial a}{\partial (a+ib)} + \frac{\partial (a^2+b^2)}{\partial b}\frac{\partial b}{\partial (a+ib)} = 2a +2b\frac{1}{i} = 2 (a-ib) = 2 z^*$