If a complex number is purely imaginary, then it must be equal to negative of its conjugate.

$\Rightarrow \frac{z-\alpha }{z+\alpha }=-\left(\overline{\frac{z-\alpha }{z+\alpha }}\right)\Rightarrow \frac{z-\alpha }{z+\alpha }=-\left(\frac{\stackrel{-}{z}-\alpha }{\stackrel{-}{z}+\alpha }\right)$

$\Rightarrow \left(z-\alpha \right)\left(\stackrel{-}{z}+\alpha \right)=\left(z+\alpha \right)\left(\stackrel{-}{z}-\alpha \right)$

$\Rightarrow z\stackrel{-}{z}+\alpha z-\alpha \stackrel{-}{z}-{\alpha }^2=-\left(z\stackrel{-}{z}-\alpha z+\alpha \stackrel{-}{z}-{\alpha }^2\right)$

$\Rightarrow {\left|z\right|}^2={\alpha }^2$

$\Rightarrow {\alpha }^2=4 \left[\because \left|z\right|=2\right]$

$\Rightarrow \alpha =\pm 2$