Unit complex numbers represent rotations in 2D. In general, multiplication by a complex number corresponds to a spiral similarity (twist and zoom) centered at the origin of the complex plane. This a two-parameter transformation.
Naively, a spiral similarity centered at the origin in 3D seems to be a four-parameter transformation; two parameters to define the axis, one scale factor and one angle. In fact this is a "double cover" because of antipodal points representing the same axis, AFAICS.
I know that representing 3D rotations with unit quaternions does NOT work like this—but is it valid to observe that a 3D spiral similarity centred at the origin has four degrees of freedom, like a quaternion? Apologies for the informality of this.