Skin depth from EM dispersion relation in conductors

[vanhees71]

This equation indicates B leading E by 45 deg. which is incorrect unless you're making an assumption that I can't fathom. B should lag by 45 deg. I, vanhees & the problem's statement all agree on this. So your error must have taken place ahead of this relation unless, again, somewhere there's an inconsistency regarding convention.

That's all correct. If you use my convention of the signs, i.e.,
$$\vec{B} \propto \exp(-\mathrm{i} \omega t).$$
Your case is for a good conductor, i.e., $\sigma \mu \omega \gg \epsilon \mu \omega^2$ or
$$\frac{\sigma}{\epsilon} \gg 1.$$
Then you have
$$k \simeq \sqrt{\frac{\sigma \mu}{\omega}} \exp(+\mathrm{i} \pi/4).$$
That means
$$B_0 \exp(-\mathrm{i} \omega t)=\sqrt{\frac{\sigma \mu}{\omega}} E_0 \exp[-\mathrm{i}(\omega t-\pi/4)].$$
This means the phase of the B field is behind that of the E field by an amount of $\pi/4$ (in the limit of good conductivity).