[Strilanc]

> why represent states as 2d complex vectors instead of quarterions? Aren't they the same thing?

Only in the sense that they are both require four real coefficients. The quaternions have a particular multiplicative structure that just doesn't apply to quantum states, so it doesn't make sense to use them.

That being said, the space of single-qubit operations is very much analogous to rotations in 3d and so is well described by quaternions. In fact, the Pauli matrices times i (iX,iY,iZ) are isomorphic to the quaternions (i,j,k). For example, iX * iY * iZ = -I.