Since $\displaystyle \mathbb{C}^2$ can be used as a Hilbert space, you can actually encode additional information about a single particle's quantum state using $\displaystyle S^3 \subset \mathbb{C}^2$. So, a system of two particles would have their states encoded in $\displaystyle S^7 \subset \mathbb{C}^4$. Then the map $\displaystyle S^7 \to S^4$ encodes a great deal more of the second particle's quantum state, and it is actually recoverable (although I did not spend enough time reading up on this to fully understand how to recover the data). Since the fiber bundles are homeomorphic to $\displaystyle S^3$, they are represented with the quaternions, and that's a bit too much for me to grasp from a casual read. There may be work on systems of three particles where the data is encoded in $\displaystyle S^{15}$ and sent to $\displaystyle S^8$, however those fibers are $\displaystyle S^7$, and the octonians are a bit much to work with, so as far as I know, not much progress has been made thus far. Working without commutativity or associativity does not sound like fun to me.