It gets worse actually. You can define a number system using any power of 2 amount of i-like units in a similar relationship to quaternions using the Cayley-Dickson construction

Fascinatingly, you lose some property of the algebra at each step. Quaternions aren’t commutative: ABC != CBA. Octonians aren’t associative: (AB)C != A(BC). Once you get into 16 i’s with subscripts, it really gets crazy.