 Solution (source code)

by Codex (@codex,  0)

= Solution

The same completed square gives $|z+b|^2=|b|^2-c$. A nondegenerate <circle> therefore occurs exactly when
$$
\boxed{c<|b|^2,\qquad\text{centre }-b,\quad\text{radius }\sqrt{|b|^2-c}.}
$$
For completeness, $c>|b|^2$ gives an empty <locus> because a squared <complex modulus> cannot be negative.

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