  We study a modification of the Higgs inflation scenario where we introduce an
extra scalar $\phi$, with mass $m$, coupled to the Ricci scalar as $g\phi^2 R$,
and mixed with the Higgs field $h$ via the Lagrangian term $\mu \phi h^2$. Both
fields participate in the inflation process in a unitary theory that predicts
values of the cosmological observables in agreement with the results from the
Planck/BICEP/Keck collaborations. In addition, by means of a $\mathcal{CP}$-odd
effective operator that couples $\phi$ to the Chern-Simons term of the
hypercharge gauge group as $f_\phi^{-1}\phi \,Y_{\mu\nu}\tilde Y^{\mu\nu}$,
maximally helical magnetic fields are produced during the last $e$-folds of
inflation. We found a window in the coupling $ f_\phi$ where these fields
survive all constraints until the electroweak phase transition, and source the
baryon asymmetry of the Universe through the Standard Model chiral anomaly.
From a phenomenological perspective, the model can solve the Standard Model
instability problem at the scale $\mathcal Q_I\simeq 10^{11}$ GeV, provided
that $\mu\lesssim m \lesssim \mathcal Q_I$, and for $m\lesssim
\mathcal{O}$(few)~TeV, the $\phi$-$h$ mixing becomes sizable while the theory
turns natural. The latter thus predicts modifications of the trilinear and
quartic couplings that could be explored at the HE-LHC, as well as at future
colliders, and allows for direct $\phi$ production at the LHC followed by decay
into $hh$. Present results from ATLAS and CMS already put (mild) bounds on the
mass of the heavy scalar as $m\gtrsim 0.55$~TeV at 95\% C.L.
