This is a tracker for a number of requests for making working with complex numbers a bit easier/faster

• Get just the normalized number z/|z| (the sgn function) (see API,ENH: Change definition of complex sign #25441, which changes the behaviour of np.sign to match the Array API and extends it to copysign)
• Get the squared modulus of a complex number z * z.conj() (instead of the slow np.abs(z)**2). square might be logical but is already defined to just do z*z... (see abs() is slow for complex, add abs2() #3994)
• A fast way to get the inverse of angle, i.e., exp(1j * a) = cos(a) + 1j * sin(a). Note that for large angle arrays, exp(1j*a) needlessly triples memory use for the input (see Have an exp for pure imaginary numbers #5625; somewhat related: sincos function, see Implement sincos() (Trac #2034) #2626; discussion moved to ENH: Create a place for "optimization" ufuncs #18483)
• Possibly, combined abs and angle and their inverse, perhaps most logical as part of providing polar to cartesian transformations (see ENH: Add to numpy simple functions for transform coordinate systems #5228)

EDIT (2021-Jun-25): if there is worry about making the numpy API too big, one option might be to make this part of numpy.lib.scimath (which I must admit I didn't know about until today; I see now that it should be np.emath - https://numpy.org/devdocs/reference/routines.emath.html). Though perhaps that should stay reserved for complex continuations real->complex like for sqrt.