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.
Usually, $\|z\|$ is defined on vectors, while $|z|$ is defined on scalars. Nevertheless, since $z$ is also a one-dimensional vector (over $\mathbb C$) or a two-dimensional real vector, we have $|z| = \|z\|$.