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First of all, never forget that including an imaginary component of an electric field is simply a useful mathematical trick we use for more conveniently accounting for phase, applicable to quantities that vary sinusoidally in time and/or space. Real-world "measurable" electric fields are all real-valued. Are you truly doing electrostatics? Or do you perhaps, have a non-zero frequency or non-zero wavenumber in your model, somewhere? If so, that would certainly explain why you have complex numbers involved in the first place. Now, the next question: What "force" are you calculating and how are you calculating it? Physical-world "measurable" forces are likewise purely real-valued. Are you attempting to model/compute: (1) a sinusoidally varying force or (2) a (for example) time-averaged force? If the former, then you would expect to find a complex force (to be interpreted as already noted, as a tool for more easily managing phase). If the latter, then expect a real-valued one. (And, of course, if you are truly doing pure electrostatics, then there is probably no reason to have employed any complex numbers in the first place. Well, unless you are (perhaps) dealing with spatially-periodic structures and handling that periodicity in terms of complex numbers to manage spatial phase. Or, I guess, if you were alternatively solving your electrostatics problems using conformal mapping methods, that would also involve complex numbers. And there could be other special electrostatic cases, I suppose, (perhaps in modeling non-linear behaviors?) where complex numbers might be of use. But anyway, if you are attempting something either subtle and/or complicated like that, perhaps you should post your model to the forum?)

Added: If you are simply asking about the elementary-level problem of a fixed static charge near a lump of unmoving isotropic homogeneous linear dielectric material, and are investigating the electrostatic force between them, then any imaginary component to the dielectric constant of the material is irrelevant. But then, there wouldn't be any imaginary component to the electric field either. But... you said that you found an imaginary component to that field, which was also as you expected. So your question is thus a bit puzzling to me.

First of all, never forget that including an imaginary component of an electric field is simply a useful mathematical trick we use for more conveniently accounting for phase, applicable to quantities that vary sinusoidally in time and/or space. Real-world "measurable" electric fields are all real-valued. Are you truly doing electrostatics? Or do you perhaps, have a non-zero frequency or non-zero wavenumber in your model, somewhere? If so, that would certainly explain why you have complex numbers involved in the first place. Now, the next question: What "force" are you calculating and how are you calculating it? Physical-world "measurable" forces are likewise purely real-valued. Are you attempting to model/compute: (1) a sinusoidally varying force or (2) a (for example) time-averaged force? If the former, then you would expect to find a complex force (to be interpreted as already noted, as a tool for more easily managing phase). If the latter, then expect a real-valued one. (And, of course, if you are truly doing pure electrostatics, then there is probably no reason to have employed any complex numbers in the first place. Well, unless you are (perhaps) dealing with spatially-periodic structures and handling that periodicity in terms of complex numbers to manage spatial phase. Or, I guess, if you were alternatively solving your electrostatics problems using conformal mapping methods, that would also involve complex numbers. And there could be other special electrostatic cases, I suppose, (perhaps in modeling non-linear behaviors?) where complex numbers might be of use. But anyway, if you are attempting something either subtle and/or complicated like that, perhaps you should post your model to the forum?)

Added: If you are simply asking about the elementary-level problem of a fixed static charge near a lump of unmoving isotropic homogeneous linear dielectric material, and are investigating the electrostatic force between them, then any imaginary component to the dielectric constant of the material is irrelevant. But then, there wouldn't be any imaginary component to the electric field either. But... you said that you found an imaginary component to that field, which was also as you expected. So your question is thus a bit puzzling to me.

Hi Robert,

Thank you very much for the detailed reply. indeed I'm attempting to simulate the quasi-static senario.

I also tried to use frequency domain study in electrostatics but still got real valued force so I'm a little confused why that's the case even though the electric field is indeed complex.