[gugagore]

I follow your connection, though I think your unit analogy is a strawman.

Do you want to give two different types to complex numbers, depending on whether a given complex number represents a point versus a transformation (an amplitwist)?

[jaen]

Ideally, yes. Depends on the context. Same way you don't use unit algebra when doing trivial everyday math. At some level of complexity, it becomes worthwhile.