= Solution
The <complex modulus> $|z-(-1-i)|$ is the <Euclidean distance> from $z$ to $-1-i$. Thus the locus is
$$
\boxed{(x+1)^2+(y+1)^2=64,}
$$
a <circle> with centre $(-1,-1)$ and <radius> $8$. Its extreme horizontal coordinates are $-9$ and $7$, and its extreme vertical coordinates are also $-9$ and $7$.
\Image[/past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2022/paper-1-q11-loci.png]
{title=The circle and its three transformed loci in Question 11, drawn with equal horizontal and vertical scales}
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