Proof

$x\cdot y:=(a,b) \cdot (c,d)$ by definition of complex numbers, the complex numbers $x,y \in \mathbb C$ are identified by pairs $x:=(a,b)$ , $y:=(c,d)$ of real numbers $a,b,c,d\in\mathbb R$ .
$=(ac - bd,~ ad + bc)$ by definition of complex multiplication.
$=(ca - db,~ da + cb)$ because multiplying real numbers is commutative.
$=(ca - db,~ cb +da)$ because adding real numbers is commutative.
$= (c,d)\cdot (a,b)$ by definition of complex multiplication.
$= y \cdot x$ by definition of complex numbers.
Therefore, the multiplication of complex numbers is commutative.
∎

Thank you to the contributors under CC BY-SA 4.0!

Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983