This affects also acoth(), asec() and acsc(). In short, everything, that's using 1/z for inversion. E.g., the acot() implemented as:

def acot(ctx, z):
    if not z:
        return ctx.pi * 0.5
    else:
        return ctx.atan(ctx.one / z)

But:

>>> mp.one/mp.mpc(cmath.infj)
mpc(real='0.0', imag='nan')
>>> fp.one/fp.mpc(cmath.infj)
-0j

On another hand, we have:

>>> mp.mpc(cmath.infj)**-1
mpc(real='0.0', imag='0.0')
>>> fp.mpc(cmath.infj)**-1
(nan+nanj)

Of course, it's yet another instance of this issue. With special rules for mixed arithmetic (e.g. real+complex as in this case), we should have -0j (or just 0 for the mp context, without signed zero).

As a workaround, we could add a branch for infinite z.