The first condition $|z+2|=|z-2|$ describes the perpendicular bisector of $(-2,0)$ and $(2,0)$, which is the imaginary axis. Write $z=iy$ with $y\in \mathbb{R}$.

Then $\frac{z+3}{z-i}=\frac{3+iy}{iy-i}=\frac{3+iy}{i(y-1)}$. Multiplying numerator and denominator by $-i$:

$\frac{(3+iy)(-i)}{y-1}=\frac{y-3i}{y-1}.$

For $\arg =\frac{\pi }{4}$, the real and imaginary parts must be equal and both positive:

$\frac{y}{y-1}=\frac{-3}{y-1}>0\Rightarrow y=-3.$

Hence $z=-3i$ and $|z{|}^2=9$.