[ziofill]

In Wirtinger calculus (https://en.m.wikipedia.org/wiki/Wirtinger_derivatives) you consider a complex variable and its conjugate as independent. This simplifies a lot of things e.g. Cauchy Riemann becomes just df/dz* = 0.

TensorFlow works this way, jax instead differentiates real and imaginary parts.

I wonder if there is a version for quaternions now.

[hbrav]

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.

[alasarmas]

I think some of the stars for your complex conjugates fell foul of HN formatting:

https://news.ycombinator.com/formatdoc