= Solution
Write $z=re^{i\theta}$ with $r>0$. On a fixed <branch of the complex logarithm> on which $\log z=\log r+i\theta$ and $\log\bar z=\log r-i\theta$, the equation becomes
$$
\log r+i\theta=i(\log r-i\theta)=\theta+i\log r.
$$
Its real and imaginary parts give the same condition $\log r=\theta$, or
$$
\boxed{r=e^\theta}.
$$
This is a <logarithmic spiral>. For the <principal complex logarithm>, it is the portion parametrized by $-\pi<\theta<\pi$ that avoids the branch cut.
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