For $z, a \in \mathbb{C}$ define the principal value of $\log z$ and hence of $z^{a}$. Hence find all solutions to (i) $z^{\mathrm{i}}=1$ (ii) $z^{\mathrm{i}}+\bar{z}^{\mathrm{i}}=2 \mathrm{i}$,

and sketch the curve $\left|z^{\mathrm{i}+1}\right|=1$.