A hypothetical particle with complex rest mass would always travel faster
than the speed of light. Such particles are called tachyons. There is a
strong chance that LENR is based on analog tachyons via metallic hydrogen
nanoparticles.

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If the rest mass m is Complex this implies that the denominator is Complex
because the total energy is observable and thus must be real. Therefore,
the quantity under the square root must be negative, which can only happen
if v is greater than c. As noted by Gregory Benford et al., special
relativity implies that tachyons, if they existed, could be used to
communicate backwards in time (see tachyonic antitelephone). Because time
travel is considered to be unphysical, tachyons are believed by physicists
either not to exist, or else to be incapable of interacting with normal
matter.
*https://en.wikipedia.org/wiki/Tachyonic_antitelephone
<https://en.wikipedia.org/wiki/Tachyonic_antitelephone>*
In quantum field theory, complex mass would induce tachyon condensation.
We know that tachyon condensation exists as demonstrated in Leif Holmlid's
experiments where metallic hydrogen produces mesons. If Holmlid is
producing mesons, it may be possible using LENR to produce a time machine.