Definition:Bounded Mapping/Complex-Valued

Definition
Let $f: S \to \C$ be a complex-valued function.

Then $f$ is bounded iff the real-valued function $\left|{f}\right|: S \to \R$ is bounded, where $\left|{f}\right|$ is the modulus of $f$.

That is, $f$ is bounded if there is a constant $K \ge 0$ such that $\left|{f \left({z}\right)}\right| \le K$ for all $z \in S$.

Also see

 * Complex Plane is Metric Space: this definition coincides with the definition of a bounded mapping to a metric space, using the standard metric on $\C$.