Can someone explain why complex analytic functions should be open mappings. A complex analytic function $f:D \to \mathbb{C}$ on some open domain $D$ can be thought of as $f:D \to \mathbb{R}^2$, $f(x,y)= (f_1(x,y),f_2(x,y))$. The Cauchy Riemann equations then tell us that the total derivative of $f$ is
$\begin{pmatrix} \partial_x f_1 & -\partial_x f_2 \\ \partial_x f_2 & \partial_x f_1 \end{pmatrix}$
whose determinant is nonzero whenever f'(z) \neq 0. Thus the inverse function theorem tells us that $f$ is an open mapping if we knew that f'(z) was never zero.
Can someone explain conceptually why open mapping should hold around a point $z$ where f'(z)=0 where the inverse function theorem itself is not enough to show openness?