Let $\left(z\right)$ represent the principal argument of the complex number $z$.

The, $\left|z\right|=3$ will represent a circle with centre $\left(0,0\right)$ and radius $3 \text{units}$

And  $\arg \left(z-1\right)-\arg \left(z+1\right)=\frac{\pi }{4}$ will represent arc of the circle with centre $\left(0,1\right)$ and radius $\sqrt{2}$, now on plotting the diagram,

We can see they are intersecting nowhere.