Power maps of a reciprocal-coordinate hyperbola ($z=t+ia/t,\mathrm{Re}(z^4)=[\mathrm{Im}(z^4){]}^2/(16a^2)-4a^2$)

Let $a>0$, $t>0$ and $z=t+ia/t$. The real and imaginary coordinates obey $XY=a$, a hyperbola. Put $s=t^2-a^2/t^2$, which is strictly increasing. Then $z^2=s+2ia$ traces a horizontal line, and $z^4=s^2-4a^2+4ias$ traces a parabola. Eliminating $s$ gives the displayed quadratic relation between real part and imaginary part. The value of $t$ fixes an orientation and determines which segment is traversed; the full real parabola is obtained as $t$ ranges over $(0,\infty )$.