Let's say we have defined the complex numbers as the set $\Bbb R imes\Bbb R$ together with addition $$(a,b)+(c,d)=(a+c,b+d)$$ and multiplication $$(a,b)\cdot(c,d)=(ac-bd,ad+bc).$$It is then possible to show that this structure is a field, which basically means that all the usual algebraic laws hold.