I was studying introductory quantum mechanics. There I came across the classic two state quantum system, the spin state. Let directions of space (physical) is denoted by x,y,z.

Spin can be represented as a two dimensional Vector. Let's take spins along direction |z > and |z-> as basis. Now other Spin states along each orthogonal direction of physical space can be represented as,

|x > = 1/sqrt(2) |z > 1/sqrt(2) |z->

|x-> = 1/sqrt(2) |z > - 1/sqrt(2) |z->

|y > = 1/sqrt(2) |z > i/sqrt(2) |z->

|y-> = 1/sqrt(2) |z > - i/sqrt(2) |z->

That's it! We can not represent any more pair of mutually orthogonal state vectors using complex number coefficients which give 1/2 inner product square with the previous states.

So if by chance, we happen to live in 4 dimensional space lets say, x,y,z,k ... how could the |k > and |k-> states be represented?

Is it a coincidence that complex numbers allow us to represent spin states living in upto 3D space and we are actually living in 3D space????