The image of a horizontal line Im(z)=pi/4 for the complex number f(z)=e^z
If $z=x+\frac\pi4i$, then$$\exp(z)=e^x\left(\frac{1+i}{\sqrt2}\right).$$Since $\{e^x\mid x\in\Bbb R\}=(0,\infty)$, $\left\{\exp\left(x+\frac\pi 4i\right)\,\middle|\,x\in\Bbb R\right\}$ is the open ray with origin in $0$ and passing through $\frac{1+i}{\sqrt2}$.