= 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. Hence
$$
\boxed{z=\pm i\text{ are simple poles};\qquad z=0,\infty\text{ are branch points}.}
$$
A 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.
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