There are few formulations of this so called "maximum principle". All of these follow from 6. Theorem 2.2.3 - Open Mapping Theorem

1. Let $D\subseteq \mathbb{C}$ be a domain, and let $f:D\to \mathbb{C}$ be holomorphic non-constant. Then $|f(z_0)|$ 1;;cannot attain a local maximum.
2. Let $D$ be a bounded domain, suppose $f:\bar{D}\to \mathbb{C}$ is continuous, $f:D\to \mathbb{C}$ is holomorphic. Then $f$ 1;;achieves its local maximum on $\delta D$
3. Let $D$ be a bounded domain in $\mathbb{C}\cong {\mathbb{R}}^{\mathbb{2}}$, $u:\bar{D}\to \mathbb{R}$ a continuous function with $u:D\to \mathbb{R}$ harmonic. Then 1;;$u$ achieves its maximum on $\delta D$.