Quaternions cover SO(3). Useful for rotations in 3d. Complex numbers cover SO(2). Useful for rotations in 2d. Geometric algebra contains both of these and a b

Quaternions cover SO(3). Useful for rotations in 3d. Complex numbers cover SO(2). Useful for rotations in 2d. Geometric algebra contains both of these and a bunch of others. Useful for generalizable operations in any number of dimensions. As an example, you might be able to apply your intuition for quaternions to spacetime geometry.