hmm. as a vector space, over the real field, there's not much difference between R^2 and C. the complex number a + bi can be seen as a linear combination of the the basis {1,i} just as a point in R^2 can be seen as a linear combination of the basis {(1,0), (0,1)}.
but vector spaces, in general, do not have a multiplication: VxV--->V that obeys the distributive law. some do, these are interesting because of that.
some examples of vector spaces WITH such a multiplication:
R[x], the vector space of all polynomials in x (if you consider only polynomials of degree 1 or less (linear polynomials and constant polynomials), this also "looks like R^2 and C", you can assign 1 = (1,0) and x = (0,1), but then the (usual) multiplication of polynomials is not closed since x^2 is not in the space).
M2[R], the vector space of all 2x2 real matrices, with matrix multipliciation. this multiplication is not commutative, AB is usually not the same as BA. it is interesting that one can actually view the complex numbers as a SUBSET of this vector space,by a+bi --->
[ a b]
[-b a]
C, the complex numbers itself.
vector spaces that have this "additional structure" are given a special name, "(associative) algebra".
of the three algebras listed above, only one, the complex numbers, possesses multiplicative inverses for every non-zero element (R[x] has inverses for constant polynomials only, and M2[R] has only has inverses for matrices with non-zero determinant). this again is somewhat special such algebras are called "division algebras". division algebras are relatively scarce in the world of mathematics, if one desires that a division algebra also be a vector space over the field of real numbers, one finds that the choice of dimensions is severely restricted (for example, there is no 3-dimensional division algebra over the reals, a fact that has vexed physicists for quite some time).
complex numbers represent a fork in the road in terms of how to view higher dimensional structures. to the left, one tries to keep the algebraic properties as much as one can, leading eventually to quaternions, octonions, and various clifford algebras. to the right, one tries to keep the higher dimensional structures the same as the reals, "only more copies", leading to hilbert spaces and differential manifolds (amongst other things).
in other words, the complex numbers have a rather unique collection of properties. they are a structure in which math is particularly nice to do. complex numbers can be viewed as numbers, as matrices, as vectors. they have nice algebraic properties, they are well-suited for solving equations in, and posses desireable spatial and geometric properties as well.