Currently there is 1 known type of electric field, it has positive(positrons and protons) and negative(electrons) charges. I would expect the charges of alternate electric fields to act the same regarding attraction and repulsion. They could hypothetically have different electric permittivities and magnetic permeabilities though.

I'm wondering if it's possible that there are other kinds of electric fields with their very own charges in existence. If they did exist, they would be very hard to detect.

Because these charges would be electrically neutral from our electric field's perspective they would essentially act like neutrinos and pass through everything. They may be able to interact with our matter gravitationally. This might be the cause of unusual phenomena like dark matter and energy.

Is it possible, given our current understanding of physics, that these alter-charges could exist?

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    $\begingroup$ Other charges do exist - the weak hypercharge and the color charge, though both of those work very differently than electric charge. $\endgroup$ – probably_someone Aug 21 '18 at 20:40
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    $\begingroup$ As far as I know, there is nothing preventing another unbroken $U(1)$ gauge group from being appended to the standard model, which would give dynamics basically identical to QED but with different charges (indeed, many "dark photon" searches concentrate on exactly this kind of model). The issue is that an unbroken $U(1)$ gauge group requires a massless dark photon, as a mass term for the photon violates gauge invariance. In order to make a massive dark photon work, you would have to break the $U(1)$ gauge symmetry by either introducing another gauge group or mixing it with the existing ones. $\endgroup$ – probably_someone Aug 21 '18 at 21:09
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    $\begingroup$ @safesphere The U(1) gauge symmetry in electromagnetism comes from symmetries found in the electromagnetic field. The U(1) gauge symmetry in the dark sector would come from symmetries found in the field corresponding to the alternate charge, which would behave identically to the electromagnetic field, but would not be the same thing, as objects might have different alternate charges than electric charges. Since the character of the gauge symmetry comes from the behavior of the Lagrangian of the field itself, there's nothing stopping you from having multiple unrelated U(1) gauge fields. $\endgroup$ – probably_someone Aug 21 '18 at 21:27
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    $\begingroup$ @safesphere I have no idea why you think that an extension to the Standard Model containing an extra massless U(1) vector boson would have no physical meaning. For example, one physical consequence would be that we can detect a new massless U(1) vector boson, that influences particles differently than the photon because its coupling strength depends on a different charge. That's literally something we can see. $\endgroup$ – probably_someone Aug 21 '18 at 21:30
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    $\begingroup$ @safesphere None of the Standard Model fields are "caused" to exist. Their existence is an axiom in the Standard Model, and each one behaves according to a certain set of (possibly-broken) gauge symmetries. I don't know where you're getting your reasoning, but it might help to clarify what, exactly, you mean by "a symmetry of nature," or to provide a reference from which you're drawing such a definition. And the current status of searches has nothing to do with this discussion; I'm arguing an extra U(1) can exist, you're arguing it can't. Neither of us is arguing that it does exist. $\endgroup$ – probably_someone Aug 21 '18 at 21:45