You are misunderstanding. A quaternion literally is a scalar + bivector (“imaginary”) rotor. We are talking about the same 4 numbers.

It lives inside a geometric algebra which also includes vectors and trivectors, but those are not part of the rotor.

If you store a rotor as a general 3D multivector, it will have 8 entries 4 of which are always 0; this could be done to simplify your code (then you only need one multivector type), but is not a good idea for efficiency of computation/transmission if you need to represent large numbers of rotations and their transmission might be a bottleneck.