I am currently tackling an optimization problem involving complex valued vectors. However the optimization is solely about finding the optimal “direction” of the vector. So any (complex-valued) scaling does not matter. To make the optimization procedure (currently gradient descent) faster and more robust I want to constraint the optimization to the complex valued unit hypersphere.
I would be interested whether there is something like complex-valued spherical coordinates so I can translate my problem into coordinates which have one degree of freedom less?
I would also be interested whether there is something like constraint gradient descent for my purpose (Maybe similar to Lagragian optimization, but in my case there is likely no closed-form solution) ?

I tried to follow the reasoning of real-valued spherical coordinates but I ran into complex valued angles and I dont know how to interprete these and properly translate from cartesian coordinates.
I also tried to normalize my target-vector after each gradient descent step but this does not increase convergence speed or robustness. And I assume to directly use the constraint in the gradient calculation or translating coordinates to reduce the degrees of freedom really is different and more promising.