= Solution
Use <polar coordinates> $z=re^{i\theta}$, where $r=|z|\geq0$. The equation sets $r=\theta$, giving an <Archimedean spiral>. The origin is excluded because the <complex argument> of zero is undefined. With the principal <complex argument> in $(-\pi,\pi]$, the locus is
$$
\boxed{z(\theta)=\theta\cos\theta+i\theta\sin\theta,\qquad0<\theta\leq\pi.}
$$
It approaches the excluded origin along the positive real axis, passes through $i\pi/2$, and ends at $-\pi$. The choice of argument matters: the convention $[0,2\pi)$ instead gives $0<\theta<2\pi$, while allowing all values of the <complex argument> gives every turn with $\theta>0$.
\Image[/past-exam-of-the-mathematical-methods-i-course-of-the-university-of-oxford/2015-spiral-and-ellipse.png]
{title=The principal-argument spiral and the rotated ellipse obtained from the quadratic form}
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