Solution
= Solution
Squaring gives
$$
u=z^2=t^2-t^{-2}+2i.
$$
Thus $\operatorname{Im}u=2$ is constant, while $d(\operatorname{Re}u)/dt=2t+2t^{-3}>0$. The trajectory is the horizontal line segment
$$
\boxed{u:\ (-15/4,2)\longrightarrow(15/4,2),\qquad\operatorname{Im}u=2.}
$$
It crosses the <imaginary axis> at $\boxed{(0,2)}$, when $t=1$, and never crosses the <real axis>. The middle panel shows the rightward direction.