Comment by meindnoch

Sure, you can say that complex numbers are 2-tuples with a special rule for multiplication (a, b) * (c, d) = (ac - bd, ad + bc). Same way you can define rationals as 2-tuples with a special rule for addition (a, b) + (c, d) = ((ad + cb) / gcd(b, d), bd / gcd(b, d)).

But I think this way you'd lose insight as to where these rules really come from. The rule for complex multiplication is the way it is, precisely because it gives you an algebra that works as if you were manipulating a quantity that squared to -1.