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Moser derived a normal form for the family of four-dimensional, quadratic,symplectic maps in 1994. This six-parameter family generalizes H\'enon'subiquitous 2D map and provides a local approximation for the dynamics of moregeneral 4D maps. We show that the bounded dynamics of Moser's family isorganized by a codimension-three bifurcation that creates four fixed points---abifurcation analogous to a doubled, saddle-center---which we call aquadfurcation.  In some sectors of parameter space a quadfurcation creates four fixed pointsfrom none, and in others it is the collision of a pair of fixed points thatre-emerge as two or possibly four. In the simplest case the dynamics is similarto the cross product of a pair of H\'enon maps, but more typically thestability of the created fixed points does not have this simple form. Up to twoof the fixed points can be doubly-elliptic and be surrounded by bubbles ofinvariant two-tori; these dominate the set of bounded orbits. The quadfurcationcan also create one or two complex-unstable (Krein) fixed points.  Special cases of the quadfurcation correspond to a pair of weakly coupledH\'enon maps near their saddle-center bifurcations. The quadfurcation alsooccurs in the creation of accelerator modes in a 4D standard map.

Elliptic Bubbles in Moser's 4D Quadratic Map: the Quadfurcation