  We study the superconducting density of states and vortex lattice of single
crystals of CaKFe$_4$As$_4$ using a scanning tunneling microscope (STM). This
material has a critical temperature of $T_c= 35\,$ K, which is one of the
highest among stoichiometric iron based superconductors (FeBSC) and is
comparable to $T_c$ found near optimal doping in other FeBSC. Using
quasi-particle interference we identify the hole sheets around the zone center
and find that two superconducting gaps open in these sheets. The scattering
centers are small defects that can be localized in the surface topography and
just produce quasiparticle interference, without suppressing the
superconducting order parameter. This shows that sign inversion is not within
hole bands, but between hole and the electron bands. Vortex core bound states
show electron-hole asymmetric bound states due to proximity of the top of one
of the hole bands to the Fermi level $E_F$. This places CaKFe$_4$As$_4$ in a
similar situation as FeSe or related materials, with a superconducting gap
$\Delta$ just a few times smaller than $E_F$. On the other hand, we also
identify locations showing strong suppression of the superconducting order
parameter. Their size is of order of the vortex core size and vortices are
pinned at these locations, leading to a disordered vortex lattice.
