[Tainnor]

C is a plane with multiplication. I would say, the fact that there is even a way of making this work is surprising (it doesn't work for any R^n, n > 2, at least not if you want a field). The particular way how we multiply complex numbers is also IMHO not self-evident (neither in the algebraic, nor in the geometric interpretation). It just turns out that doing it this way gives us really nice properties.

[kazinator]

> how we multiply complex numbers is also IMHO not self-evident

Multiplication of complex number in the x + iy form treats them exactly like any other binomial factors, using the "FOIL" rule:

                       F    O     I     L

   (a + bi)(c + di) =  ac + adi + bci + bdi^2

                    =  ac + (ad + bc)i - bd

                    =  ac - bd + (ad + bc)i

I don't know about self-evident; you pretty much have to do it that way. If you hand (a + bi)(c + di) to someone who knows basic algebra, but has no idea what i is, they will come up with the first FOIL expression above; after that, we have to know that i^2 is -1.

> nor in the geometric interpretation.

That isn't self-evident, but when you gemoetrically work out what the multiplication is doing, it's just "add the arguments (angles), multiply the moduli (distances from origin)".

Very simple!

[Tainnor]

> you pretty much have to do it that way

You have to do it that way because it's the only way to get a field out of R^2, but I don't think it's at all self-evident. The "we have to know that i^2 is -1" is doing some heavy lifting in your explanation. In the context of "complex numbers are just a 2d plane" there's no a priory reason why you'd think that (0,1)*(0,1) has any particular meaning, let alone that it's equal to (-1,0).

If we interpret vectors in R^n as dislocation (or as forces, or...), then vector addition behaves exactly as one would expect. I think with enough motivation a bright school kid could come up with the formula for vector addition. And it works for every dimension.

Vector multiplication only works for 2d (ignoring the less well behaved cross product that also only works in certain dimensions) and I don't think you'd come up with it naturally.

Also, historically, the 2d plane isn't how complex numbers were discovered. They first appeared as weird quantities that made solving (real) cubic equations simpler but were regarded as just temporary artifacts. The realisation that they can be identified with R^2 came much later.