I'm trying to calculate regular values of the map $f:\mathbb{C}^n\rightarrow \mathbb{R}$ given by $(z_1,\ldots,z_n)\mapsto \sum|z_i|^2$.
I know that for the map $g:\mathbb{R}^n\rightarrow \mathbb{R}$ given by $(x_1,\ldots,x_n)\mapsto \sum|x_i|^2$, the differential is given by the Jacobian with entries $dg/dx_j=2x_j$. Therefore the derivative is surjective for any $g(x_1,\ldots,x_n)=y>0$ and hence the regular values are all $y>0$.
I'm not sure how to do this for the complex case, if we naively consider $\mathbb{C^n}$ as $\mathbb{R^{2n}}$ do we not lose the complex structure?