Solution
= Solution
Put $z=x+iy$ and $z^*=x-iy$. The first two squared terms are <complex conjugates>, so their sum is $2[(x-2)^2-(y+1)^2]$. Also $zz^*=x^2+y^2$. The equation therefore reduces to
$$
2[(x-2)^2-(y+1)^2]+2(x^2+y^2)+6
=4(x^2-2x-y+3)=0.
$$
Thus the <locus> is the upward-opening <parabola>
$$
\boxed{y=(x-1)^2+2,}
$$
with vertex $1+2i$ and axis $x=1$ in the <Argand diagram>.