Are quantum fields such as electron field fundamental?

Within the Standard model of particle physics, the most general experimentally verified model of fundamental physics (excluding gravity), there are quantum fields, such as the electron-positron field, that are truly fundamental. Other fundamental fields within the Standard model include the tau, muon, muon- tau- and electron-neutrino fields, quark fields, gauge boson fields, and, finally, the Higgs field. Any directly observed particle to date can be explained as a composite of these fields.

Note, however, two things. First, the fact that one field can decay into another does not mean that the first field is more fundamental than the second; it simply means there is an interaction channel between the two and that the decay was thermodynamically favorable. For instance, the tau particle has a decay time of a fraction of a picosecond, and it decays into various other particles through the (electro)weak interaction - but that does not mean that the tau field is not fundamental.

On the other hand, the second thing to note is that we do not believe the Standard model to be the final story about fundamental physics. Namely quantized gravity, dark matter, the inflation scenario, or neutrino oscillations are not implemented in the current version of the Standard model. At this point there seem to be more than one way how to do that, and only new experimental results can guide us. But back to your question about the inflaton - since new elements and mechanisms are needed in the Standard model to obtain an inflationary period in the Early Universe, it may well turn out that some of the fields in the current Standard model are not fundamental and only emergent composites (even though, as noted above, it is not strictly necessary for all of them).

Quantum fields can exchange energy. So for example when an electron and a positron annihilate the energy that was in the electron/positron field is transferred to the photon field. The result is that one electron and one positron disappear and two photons are created. The fields themselves are still both present - only the energy in the fields has changed.

So the fields are not decaying into each other. They just exchange energy. At the end of inflation the inflaton field transferred its energy into other quantum fields. That annihilates the particles in the inflation field and creates particles in the other fields.

I was about to say that the inflaton field doesn't disappear after it has transferred energy to the other fields, but we need to exercise some caution because we don't know what the inflaton field was so we can't say what happened to it at the end of inflation.