  Plasmon-polariton modes in two-dimensional electron gases have a dual
field-matter nature that endows them with unusual properties when electrical
conductivity exceeds a certain threshold set by the speed of light. In this
regime plasmons display an interesting relation with tachyons, the hypothetical
faster-than-light particles. While not directly observable, tachyons directly
impact properties of plasmon modes. Namely, in the ``tachyon'' regime, plasmon
resonances remain sharp even when the carrier collision rate $\gamma$ exceeds
plasmon resonance frequency. Resonances feature a recurrent behavior as
$\gamma$ increases, first broadening and then narrowing and acquiring
asymmetric non-Lorentzian lineshapes with power-law tails extending into the
tachyon continuum $\omega>ck$. This unusual behavior can be linked to the
properties of tachyon poles located beneath $\omega>ck$ branch cuts in the
complex $\omega$ plane: as $\gamma$ grows, tachyon poles approach the light
cone and hybridize with plasmons. Narrow resonances persisting for
$\gamma>\omega$, along with the unusual density and temperature dependence of
resonance frequencies, provide clear signatures of the tachyon regime.
