= Solution
For a nonzero <complex number>, write $z=re^{i\theta}$ with $r=|z|>0$ and principal <complex argument> $-\pi<\theta\le\pi$. The <principal complex logarithm> and corresponding <complex exponentiation> are
$$
\boxed{\operatorname{Log}z=\log r+i\theta,\qquad z^a=\exp(a\operatorname{Log}z).}
$$
The value at zero is not defined by this formula. The choice of <complex argument> matters: these are single-valued principal powers, rather than all values of a multivalued <complex logarithm>.
For the requested sketch, $(1+i)\operatorname{Log}z=(\log r-\theta)+i(\log r+\theta)$. Its <complex exponential> has <complex modulus> $e^{\log r-\theta}$, so
$$
\boxed{|z^{1+i}|=1\quad\Longleftrightarrow\quad r=e^\theta,\qquad-\pi<\theta\le\pi.}
$$
This is one turn of a <logarithmic spiral>, with increasing radius as the polar angle increases. It passes through $1$ at $\theta=0$, approaches the excluded endpoint $-e^{-\pi}$ from below the negative real axis, and ends at the included endpoint $-e^\pi$ from above that axis. The endpoint distinction follows from the chosen principal <complex argument>.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2011/ia/paper-1-principal-spiral.png]
{title=One turn of the principal-power logarithmic spiral, with open inner and closed outer endpoints}
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