= Solution
By <complex exponentiation>, a value of $z^i$ has the form
$$
\exp\left(i\bigl(\log|z|+i(\arg z+2\pi k)\bigr)\right).
$$
For this to equal one, its exponent must be $2\pi in$ for some $n\in\mathbb Z$. Hence a logarithm of $z$ must equal $2\pi n$, so $z$ is positive real and
$$
\boxed{z=e^{2\pi n},\qquad n\in\mathbb Z}.
$$
With the principal branch this is the same list; in the multivalued convention these are precisely the $z$ for which one value of $z^i$ equals one.
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