Given a causal LTI system with a transfer function, determine if the system is an all-pass filter
In an all-pass filter we should have $$H(z)H^*(1/z^*)=c^2$$
which means that the poles and zeros of the system $H^*(1/z^*)$ cancel the zeros and poles of $H(z)$. 
As a result, an all-pass filter has conjugate-reciprocal pole-zero pairs. For this system:

 

$$H^*(1/z^*)=\frac{4z^2+1}{-z^2+4}$$
So 
$$H(z)H^*(1/z^*)\neq c^2$$
You can also see that the poles and zeros are not conjugate and reciprocal: $\frac{1}{\left(\pm2j\right)^*}\neq\pm\frac{1}{2}$