The problem with generalizing this to quarternions is that the conjugation operation for quaternions can be expressed using arithmetic operations on the quatern

The problem with generalizing this to quarternions is that the conjugation operation for quaternions can be expressed using arithmetic operations on the quaternion: q* = -0.5(q + iqi + jqj + kqk) So the analogy to complex analysis where we'd talk of z and z as independent doesn't work anymore - since we can write q* as an 'analytic' function of q. It's not surprising you'd need something different though, since (q, q*) is only two variables and quaternions are 4-dimensional. I don't know a lot about quaternions, but Penrose introduces them in The Road to Reality and says (roughly) "yeah, they don't have the nice analytic-function properties that complex numbers have" and seems to kinda leave it at that. If anyone knows more and wants to reduce my ignorance, I'd be grateful.