Question: Is there any difference between the absolute values operators $|z|$ and $\|z\|$ where $z=a+ib$?

Answer: 
Usually, **no**. But if you see both notation used in the same discussion, it is possible that the author intended to define $\|z\|$ to be a different norm; in that case it will strictly depend on context.
I have only seen $|\cdot |$ used in the context of complex numbers. If you think of a complex number as a real planar vector, the planar vector's $\|\cdot \|$ is the same as the corresponding complex number's $|\cdot |$.