Is there a proof for the maximum principle without the Cauchy integral theorem?

Question

All the theorems about holomorphic functions seem to rely on the Cauchy integral theorem: Liouvilles theorem about bounded whole functions, the maximum principle, the open mapping theorem for holomorphic functions, Riemanns theorem about removabe singularities, etc. I see that the Cauchy integral theorem is powerful but are there other ways to prove one of these theorems that do not directly or implicitely use the Cauchy integral theorem?

Answer 1

Alternatively there's a topological approach, all the results rely in a topological index which is used to obtain the winding number of a curve around a point.

The reference is Topological Analysis by Gordon Thomas Whyburn.