Cardinality has nothing to do with this discussion. Quarternions have different structure than real numbers. Over every set, you can find any (reasonable) struc

Cardinality has nothing to do with this discussion. Quarternions have different structure than real numbers. Over every set, you can find any (reasonable) structure you want, but that's not a very interesting question. For example Q and Z have the same cardinality but they look nothing like each other other than the fact that Q is analogous to the fraction field of Euclidean Domain Z.