> Why do you sum up coordinates?

People use sums to denote vector components, too. It's common to see ai + bj + ck (pretend there are hats on i, j, and k) to mean the vector [a b c].

So with complex numbers, you can think of it as a vector aj + bi where j and i are basis vectors, but we call j "1", because complex numbers have a notion of multiplication, and in that notion, j acts like 1, in that zj = jz = z for any complex number z.

Nothing bad happens if you write all complex numbers a + bi as vectors or tuples [a b]. It's just less convenient.