Find the locus of points for:
$$
Re(z^{2}) = 4
$$
$$
Let z=x+iy
$$
$$
Re((x+iy)^{2}) = 4
$$
$$
Re(x^{2}+2ixy-y^{2}) = 4
$$
$$
x^{2} - y^{2} = 4
$$
$$
\frac{x^{2}}{4} - \frac{y^{2}}{4} = 1
$$
Therefore, the locus of z is a hyperbolic curve with $\frac{x^{2}}{4} - \frac{y^{2}}{4} = 1$
