Perhaps it represents a harmonically oscillating charge value.

Dear Sir,

the complex permittivity is a value for a polarisation loss of a complex material value.

This is a oscillating effect of a material.

Yours Sincerely

I do not think there is a concept of complex charge. Only to deal problems of conduction in a.c. fields and dielectric material in a.c field, one defines complex conductivity, complex permittivity. In optical case where electromagnetic wave is incident on a material, even refractive index is considered as complex quantity. But no where charge is considered to be complex.

I concur with Dr. Kumar's previous answer. In classical electromagnetics, complex permittivities (DC permittivity + frequency dependent loss tangent), complex refraction angles (which represents evanescent surface waves that are generated during total internal reflection: in fact frustrated total internal reflection) and complex conductivities (again frequency dependence) exist. But I have never heard of complex charges.

I agree with Dr Kumar and Mr Manish. The EM your referring, is not a charge. It is conceptually outcome of accelerated or decelerated charge. Charge is electrical energy and EM is an electrical energy converted into wave and it does not remain any way in the form of charge.

I also agree with Drs.  A. Kumar, M. Nair and Amipara Manilal D. Conductivity can be considered to be complex but  not the charge.

Thanks to all who commented on the issue. 

I am not able to get into details here why I have such a question, but a complex charge density appears somehow in my QFT calculations, and I have one thought why is could be not so incorrect as it seems at the first glance (but it is not a proof or anything just a thought).

The presence of complex conductivity naturally brings the complex current....but the current is complex what then happens with the charge density from the charge conservation law? shouldn't it become (formally) complex as well?

Charge is a scalar quantity and hence it cannot become complex

What if the value of the charge is oscillating harmonically?

Harmonically oscillating charge meaning flow of charge and it is current. Still charge remains scalar

Charge density is NOT a scalar, it is temporal (0th) component of a 4-vector, that's first. 

Secondly. both a scalar and a 4-vector can perfectly be complex, no restrictions on that.

Dear Ignat,

Let me try to explain some points respect to your interesting question:

1.       Electric charge is one Lorentz scalar while density of charge is not. In fact, it is the time component of a 4-current density.

2.       It is true that Minkowski metric can be written in Euclidean form just using complex metric where the electric field or its induction can be represented as complex magnitudes.

3.       In such Euclidean complex representation you can apply the Gauss equation to obtain a complex electric charge. Notice that this is just a geometrical representation due to have a complex metric and no new physical meaning is behind.

4.       The main physical meaning of the electric charge is that it is a conserved quantity into Maxwell equations and in the whole electrodynamics. In fact, it appears as a Noether current associated to the abelian U(1) group in QED. Notice that this conservation cannot be never a complex quantity with a real and imaginary part because it would needed to have two parameters to conserve.

5.       Thus you can represent the electric charge as pure imaginary number or a real one depending of the geometrical representation that you use, i.e. pseudo-Euclidean or Euclidean. But the physical meaning must be always the same.

6.       Respect to complex conductivity, permittivity, permeability, susceptibility or so on. This is a very different story. All of them are due to the application of an external oscillation electromagnetic on a material with different physical properties: metal, insulator, semiconductor, ferromagnetic, ferroelectric, antiferromagnetic, superconductor, etc… which have very different dispersive equations than the ones of the vacuum. Therefore it is possible to find a meaning associated to the pure complex component as losses, but in any case the magnetic component never can be a physical observable in such a case. The main analogy is the concept of impedance instead of resistance when you work with electric circuits for ac currents

   I hope that this can helps.