  We report inelastic light scattering results on the stoichiometric and fully
ordered superconductor CaKFe$_4$As$_2$ as a function of temperature and light
polarization. In the energy range between 10 and 315 cm$^{-1}$ (1.24 and 39.1
meV) we observe the particle-hole continuum above and below the superconducting
transition temperature $T_c$ and 7 of the 8 Raman active phonons. The main
focus is placed on the analysis of the electronic excitations. Below $T_c$ all
three symmetries projected with in-plane polarizations display a redistribution
of spectral weight characteristic for superconductivity. The energies of the
pair-breaking peaks in $A_{1g}$ and $B_{2g}$ symmetry are in approximate
agreement with the results from photoemission studies. In $B_{1g}$ symmetry the
difference between normal and superconducting state is most pronounced, and the
feature is shifted downwards with respect to those in $A_{1g}$ and $B_{2g}$
symmetry. The maximum peaking at 134 cm$^{-1}$ (16.6 meV) has a substructure on
the high-energy side. We interpret the peak at 134 cm$^{-1}$ in terms of a
collective Bardasis-Schrieffer (BS) mode and the substructure as a remainder of
the pair-breaking feature on the electron bands. There is a very weak peak at
50 cm$^{-1}$ (6.2 meV) which is tentatively assigned to another BS mode.
