Let $f: \mathbb{C}^d \rightarrow \mathbb{C}^d$ be a function such that there is a $c > 0$ with $$ |\langle f(x) - f(y),x-y \rangle| \leq c \langle x-y,x-y \rangle $$ for all $x, \; y \in \mathbb{C}^d$, where $\langle \cdot , \cdot \rangle$ denotes the Hermitian product.
Does this imply that $f$ is Lipschitz continuous?