Relationship between complex number and vectors

The set of complex numbers $\mathbb{C}$ is like the set of two dimensional real vectors $\mathbb{R}^2$ in the sense that you have a one-to-one mapping from the one into the other, except it has special rules for addition and multiplication, making it into a very special field.

Generalization to higher dimensions than 2 does not hold, except somehow for the case of the quaternions.

Addition is the same but multiplication is different. One way to look at complex multiplication is that it is like a rotation in $\mathbb{R}^2$.

There is a lot of ground to cover here, so I will give a reference. A very beautiful book that spends a lot of time on a detailed explanation and comparison of the two is Needham's "Visual Complex Analysis". It is very difficult to match the explanations there in lucidity, pedagogy and geometric intuition. It is definitely one of the best mathematical prose I have read.