[ithinkso]

If you want the equation

x + 1 = 0

to have a solution, you need to invent negative numbers, now if you want the equation

x^2 + 1 = 0

to have a solution, you need to invent complex numbers and 'i'. (Also, complex numbers, turns out, are enough for higher powers as well)

The line and plane are just convenient representations of R and C but there is nothing inherently profound about them, in my opinion

[ajkjk]

This is one take, and one that has become very popular, but it's not necessarily the only take. In particular it presupposes that your number-like indeterminates can be both (a) multiplied and (b) added to numbers (and, implicitly, divided). Naturally the solution has to be a division algebra. If instead you asked the question "for what values of O would O^2 (v) = -v", or even just O^4 (v) = v, then you would be more content having the answer live in a different space, of operators on vectors rather than vectors themselves, instead of in a field extension of the present space. Of course they are basically isomorphic but I think the alternate interpretations are useful to keep in mind so that we don't accidentally assume our way into a box of our own making.

edit: I should add, by O^2 I mean O ∘ O, so there's no definition of "multiplication" on these necessarily, just composition.