It is straightforward to extend the notion of a 2D vector in the Cartesian $x-y$ plane to 3D $(x,y,z)$ or to any dimension.
Sometimes it is useful to express vectors in the complex plane, where the 2D vector has a nice compact expression:
$$e^{i\theta}=i\sin(\theta)+\cos(\theta)$$
But how do you extend this to 3 or more D? I would guess you just keep adding more cosines of different angles to the real part, but then how does this change the LHS?
$$e^{i\theta+...?}=i\sin(\theta)+\cos(\theta)+\cos(\phi)$$