Can a particle have an imaginary mass?

Mathematically, yes. Physically, no. Tachyons are a sign of an unstable theory and need to be dealt with. tachyons are these weird particles which move faster than the speed of light. Special relativity tells us that mass tends to infinity as an object's velocity tends towards light speed i.e. $$m = frac{m_0}{sqrt{1-frac{v^2}{c^2}}},$$ which as $vrightarrow c, m rightarrow infty.$ But if $v^2 > c^2, m sim -i m_0.$ When you plug this into the energy mass condition, $E = sqrt{m^2 c^4 + p^2 c^2}$, you get negative energies. (This is the sign of an instability. In fact, in Minkowski spacetime, this is not allowed.) What this means is that as the ratio $v^2/c^2 > 0$ increases, the faster it goes, the more mass this particle loses. In theory this is unbounded from below, and if there is no mechanism to prevent this particle from existing, your theory will be plagued with problems such as the existence of particles with $m^2 = -infty.$ So tachyons are not a sign of a healthy theory. In fact, this is also why bosonic string theory is plagued with problems - the vacuum state is tachyonic and one needs to fix the number of dimensions to 26 in order for this tachyon to be removed the theory. (that's not exactly the reason why you fix d = 26 in the bosonic string theory but it also seems to deal with the tachyon problem.)

But otherwise, the tachyon is very interesting from the perspective of causality as well. Particles are causal if they travel within the light cone. Tachyons do not travel inside the light cone so what this means is that $ds^2 = +1$ for the tachyon which means that it is a spacelike quantity and can travel backwards in time. (Hence, if you go faster than the speed of light, you go backwards in time!)

OK so if the math gives, why not reinterpret what negative Values for E mean? Space-Time contracts near high energies, but is flat overall, so it must expand as well. I'm not saying Tachyons are the Dark Matter we're looking for, unless that's exactly the case, and negative values for E could be interpreted in that manner. Near a concentration of negative energy, time would speed up, making every particle seem to move faster to an outside observer, and it would push matter away from its center, and forcing light to take a longer path, redshifting it. If these hypothetical imaginary mass particles existed, and they were faster than light, this would make it hard for them to concentrate in one region. BUT they themselves would also be repelled by positive energy concentrations, which have traditional mass, and they would rebound around in the megastructures of the cosmos, the matter herding them in but being pushed away by them at the same time, until the megastructures of the cosmos became so weak that they couldn't serve as a barrier anymore, and Spacetime then expands so fast that Milkdromeda can't see anyone else out there anymore. There wouldn't be any Big Rip, however, because the negative energy would still be held at bay by the matter concentration.

There needn't be a causality paradox here, either. The hypothetical imaginary mass particles themselves would only 'interact' with the curvature of spacetime, not with eachother, nor with "regular" matter. They also wouldn't concentrate into white holes, the mathematical anti-black-holes, because there's no force sticking them together; the mass/density needed to herd together enough negative energy to create a white hole would be a black hole of at least counterequal Energy(super hypothetically). No force in the universe could do that because it would be like accelerating the black hole to light speed to get it close enough to the negative energy.

Of course, all of that makes it pretty much impossible to prove - or disprove- anything I've claimed about these hypothetical particles. So, they're not even hypothetical, but purely... imaginary

Crickets

Virtual particles can have imaginary mass, so to the extent that virtual particles are real (as opposed to being just a term in a perturbation series): Yes.

Consider electron scattering off nucleons in the lab frame: A beam with energy $E$ scatters by $theta$ degrees, with a final energy of $E'$ is mediated by the exchange of a virtual photon with squared four momentum (for $E>>m$):

$$ Q^2 = EE'sin{frac{theta}{2}} > 0 $$

which corresponds to an imaginary mass. This is just the kinematics of taking 2 ultra-relativistic electron four momenta and subtracting them (the so-called $t$ channel):

$$ q_{mu} = k_{mu} - k'_{mu}$$

It is tachyon-like in the sense that the end points of the exchange are space-like. You often hear: "the electron emits a virtual photon which is then absorbed by the proton". That statement leaves out "the proton (Parton) emits a virtual photon that is absorbed by the electron".

In the old Time-ordered perturbation theory, one needed to address each process separately. The manifestly covariant Feynman diagram includes both, hence one just says "they exchange a virtual photon".