Let $\mathbb{C}$ be the D372: Set of complex numbers.
Let $+$ be the D606: Real addition operation.
Let $\cdot$ be the D610: Real multiplication operation.
The complex multiplication operation is the D554: Binary operation
\begin{equation}
\mathbb{C} \times \mathbb{C} \to \mathbb{C}, \quad
((a, b), (c, d)) \mapsto (a c - b d, a d + b c)
\end{equation}