  We obtain a complete normal form for models of real analytic uniformly
$2$-nondegenerate CR hypersurfaces in $\mathbb{C}^4$, and present a detailed
study of their local invariants. The normal form illustrates that
$2$-nondegenerate models in $\mathbb{C}^4$ comprise a moduli space
parameterized by two univariate holomorphic functions, which is in sharp
contrast to the well known Levi-nondegenerate setting and the more recently
discovered behavior of $2$-nondegenerate structures in $\mathbb{C}^3$. In
further contrast to these previously studied settings, we demonstrate that not
all $2$-nondegenerate structures in $\mathbb{C}^4$ arise as perturbations of
homogeneous models. We derive defining equations for the homogeneous models, a
set of $9$ structures, and find explicit formulas for their infinitesimal
symmetries.
