= Solution
With $z=x+iy$, the expression inside the <modulus> is $x+3i(y-1)$. Squaring its nonnegative <modulus> therefore gives
$$
\boxed{x^2+9(y-1)^2=4,\qquad\text{or}\qquad\frac{x^2}{4}+\frac{(y-1)^2}{4/9}=1.}
$$
The <locus> is an <ellipse> in the <complex plane>, centred at $(0,1)$, with horizontal semiaxis $2$ and vertical semiaxis $2/3$. Its vertices are $(\pm2,1)$ and $(0,1\pm2/3)$.
\Image[/past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2019/paper-1-2019-q11-ellipse.png]
{title=Ellipse obtained from the complex modulus condition}
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