I have explored the basis for complex numbers. I found that complex numbers really are related to ordered pairs in set theory. The number i is in fact an ordered pair $(0,1)$ and multiplication of ordered pairs follows (a,b) * (c,d) = (ac-bd,ad+bc) so that $$i^2=(0,1)^2=(0,1)*(0,1)=(-1, 0)=-1.$$
However, I can't find any explanation for this definition of ordered pair multiplication.
How is (a,b) * (c,d) = (ac-bd,ad+bc)
Once upon a time, people started doing arithmetic with numbers that involved $i$ subject to the rule $i^2=-1$. So when they multiplied $a+bi$ times $c+di$, they got $ac+adi+bic+bidi=(ac-bd)+(ad+bc)i$ (where the minus sign comes from $i^2=-1$). But their rules for doing arithmetic didn't provide an answer for the obvious question "What exactly is this $i$?" If asked this question they would answer (as some people allegedly do nowadays when asked how to interpret quantum mechanics) "Shut up and compute."
Later, people noticed that the complex numbers correspond to ordered pairs of real numbers, with $a+bi$ corresponding to $(a,b)$. [So you could even plot complex numbers as points in the plane, but that's not relevant to he present question.] So they got the bright idea of saying that the complex number $a+bi$ is the ordered pair $(a,b)$, where $a$ and $b$ are real. Now they had an answer to "What is $i$?", namely $i=0+1i=(0,1)$.
All the old algebraic facts about complex numbers could be translated in terms of this new view of what complex numbers are. Instead of writing $(a+bi)\cdot(c+di)=(ac-bd)+(ad+bc)i$, they would write $(a,b)\cdot(c,d)=(ac-bd,ad+bc)$.
They could cover up their trail by never mentioning all that stuff about $i$ and just writing definitions that say "A complex number is an ordered pair of real numbers" and "You add complex numbers by $(a,b)+(c,d)=(a+c,b+d)$" and "You multiply them by $(a,b)\cdot(c,d)=(ac-bd,ad+bc)$." They proved theorems about these, and they lived happily ever after, or at least until somebody like you came along and asked how they made up those formulas.
