Minimum modulus principle - counterexample when assumption that function is non-zero on a domain is removed

Question

Minimum modulus principle:

Let $f$ be a holomorphic function on open and connected set $D$, such that for all $z \in D : f(z) \neq 0 $. If there exists local minimum of $|f|$ then $f$ is constant.

Assumption that $f$ is non-zero on $D$ allows us to use Maximum modulus principle to prove it. I'm interested in knowing if there is a counterexample for version of this theorem with the assumption of $f$ being non-zero removed.

Answer 1

Any function which is zero at a point in the domain is a counter example.

For example $0$ is a minimum for $|f|$ when $f(z)=z$ in any domain containing $0$.