Solution
= Solution
For arbitrary complex $b$ and real $c$, complete the <complex modulus> square:
$$
zz^*+bz^*+b^*z+c=|z+b|^2+c-|b|^2.
$$
The <locus> is a single point exactly when
$$
\boxed{c=|b|^2,\qquad z=-b.}
$$
= Solution
For arbitrary complex $b$ and real $c$, complete the <complex modulus> square:
$$
zz^*+bz^*+b^*z+c=|z+b|^2+c-|b|^2.
$$
The <locus> is a single point exactly when
$$
\boxed{c=|b|^2,\qquad z=-b.}
$$