 Solution

The factor $z^{-1/4}$ has algebraic branch points at zero and infinity. Near zero the leading behaviour is $z^{-1/4}$; with $w=1/z$, the behaviour near infinity is $g(1/w)\sim w^{9/4}$. These are four-sheeted branch points.

The polynomial denominator vanishes at $z=\pm i$, with nonzero derivative $2z$, and the fractional factor is finite and nonzero near either point on any chosen local branch. HenceA single-valued holomorphic branch is obtained by choosing a branch cut joining zero to infinity. Points along the chosen cut mark its discontinuity, rather than additional isolated poles.