Is there any difference between the absolute values operators $|z|$ and $\|z\|$ where $z=a+ib$?
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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.
Willie Wong
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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 $|z| = \|z\|$.
Johannes Kloos
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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 $|\cdot |$.
ncmathsadist
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I'm pretty sure I've seen $|\cdot|$ commonly used for vectors in introductory calculus textbooks. – Erick Wong Jun 14 '12 at 18:46
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This is sometimes done. – ncmathsadist Jun 14 '12 at 22:12