  Employing a combination of symmetry analysis, low-energy modeling, and ab
initio simulations, we predict the presence of magnetic-field-induced Weyl
points close to the Fermi level in CaKFe$_4$As$_4$. Depending on the relative
strengths of the magnetic field and of the spin-orbit coupling, the Weyl
fermions can carry a topological charge of $\pm1$ or $\pm2$, making
CaKFe$_4$As$_4$ a rare realization of a double-Weyl semimetal. We further
predict experimental manifestations of these Weyl points, both in bulk
properties, such as the anomalous Hall effect, and in surface properties, such
as the emergence of prominent Fermi arcs. Because CaKFe$_4$As$_4$ displays
unconventional fully-gapped superconductivity below 30 K, our findings open a
novel route to investigate the interplay between superconductivity and Weyl
fermions.
