The speed of light in a material is defined as $c = \frac{1}{\sqrt{\epsilon \mu}}$. There are metamaterials with negative permittivity $\epsilon < 0$ and permeability $\mu < 0$ at the same time. This leads to a negative refractive index of these materials.

But do (meta-) materials exist with only negative $\epsilon < 0$ and positive $\mu > 0$ or vice versa? This would lead to a complex speed of light inside such materials.

What would be the consequences of a complex speed of light? Could particles reach unlimited speed inside these materials? Would there still be Cherenkov radiation?