A complete normal form for everywhere Levi degenerate hypersurfaces in $\mathbb C^{3}$

$2$-nondegenerate real hypersurfaces in complex manifolds play an important role in CR-geometry and the theory of Hermitian Symmetric Domains. In this paper, we construct a complete convergent normal form for everywhere $2$-nondegenerate real-analytic hypersurfaces in complex $3$-space. We do so by developing the homological approach of Chern-Moser in the $2$-nondegenerate setting. This seems to be the first such construction for hypersurfaces of infinite Catlin multitype. Our approach is based on using a rational (nonpolynomial) model for everywhere $2$-nondegenerate hypersurfaces, which is the local realization due to Fels-Kaup of the well known tube over the light cone. As an application, we obtain, in the spirit of Chern-Moser theory, a criterion for the local sphericity (i.e. local equivalence to the model) for a $2$-nondegenerate hypersurface in terms of its normal form. As another application, we obtain an explicit description of the moduli space of everywhere $2$-nondegenerate hypersurfaces.