The overall picture is the following:
Reinterpreting nonrelativistic scattering theory in terms of quantum field theory one finds that the finite poles of the renormalized propagator are the masses of the asymptotic particle states (corresponding to the bound states in the nonrelativistic theory) of the theory with the appropriate quantum numbers. Since quarks are confined, there are no such asymptotic states for the quark field, hence the renormalized quark propagator should have no finite poles. But the concept of asymptotic states is appropriate only at zero (or sufficiently low) temperatures.
At higher temperatures the role of the asymptotic states is taken instead by quasiparticle states. In quantum field theory, these correspond to representations of the observable algebra different from the vacuum representation. Here the spectrum of the Hamiltonian is not bounded below, and one has instead to satisfy the so-called KMS condition. This changes everything.
The representation becomes temperature dependent, and at sufficiently high temperature a phase transition is observed numerically that changes the quasiparticle spectrum and results in quasiparticles with the quantum numbers of quarks. The corresponding 2-point functions have complex poles describing the mass and lifetime of the quasiparticles in a similar way as, in the nonrelativistic case, complex poles of the analytically continued resolvent of a Hamiltonian describes energy and lifetime of unstable particles and resonances.
In the literature, all these things look somewhat nebulous since there is no clear common language mediating between the relativistic and the nonrelativistic case, and everything is argued in terms of analogies.