I am trying to visualize the parametric plot in $\mathbb{C}$ of the curve $\gamma$ defined for $t\in[-\infty,\infty]$ as $$\gamma(t)=\exp\left(-t^{2}+\frac{t}{\sqrt{1+t^2}}i\right).$$
I was wondering if there is a good way to think about this beyond basic intuition like it should be something periodic or a spiral thanks to Euler's formula, or that for large absolute values of $t$ the curve will approach the origin.