Proof of the proposition (Morse function in $\mathbb{C}$)

Question

I was trying to prove the Proposition asked by OP. I was thinking to apply some version of inverse function theorem as proved in the Morse Lemma. But I am not able to do it. Can you please give me some hints to proceed.

Answer 1

The proof can be done by using the special case of the Hadamrad Lemma ($n=2)$.

Answer 2

I'm going to assume that the function $f$ in the proposition is assumed holomorphic. Given that, a proof of the complex morse lemma is here:

https://en.wikipedia.org/wiki/Method_of_steepest_descent#Complex_Morse_Lemma

and a google search will lead you to other similar versions.