Every complex number axiom system, that I have ever seen, contains one of these two statements: 1) Exists a complex number $i$ : $i^2 = -1$; or 2) For any complex numbers $z_1=a+bi, z_2=c+di$ their product $z_1 \cdot z_2=(ac-bd)+(ab+cd)i$.
All that looks very unreasonable, taken from nowhere. I can't understand, why so strangely constructed set generalizes the real numbers and even keeps linearity, just as vectors (i.e. $(a+bi)+(c+di)=(a+b)+(c+d)i$ and $c(a+bi)=(ca)+(cb)i$). Of course, it's not just a coincidence!
What stands behind all this stuff and where does it come from? Why are complex numbers usually defined like that and what is a more clear way?
P.S. The question is not about how do complex numbers work, but why do they work and why does their particular model describe the most natural extension of real numbers.
I'm very curious about your use of the word "unreasonable". When your teacher told you that you were going to start using a number that you would write as $\sqrt 2$, and that
$$ (1 + 5 \sqrt 2) + (3 - 4 \sqrt 2) = (1+3) + (5-4)\sqrt 2 = 4 + 1\sqrt 2,$$
just how reasonable / unreasonable did you find that?
And to take it one step further, what do you think of the way that $ (1 + 5 \sqrt 2) \times (3 - 4 \sqrt 2)$ was calculated?