Moser derived a normal form for the family of four-dimensional (4d), quadratic, symplectic maps in 1994. This six-parameter family generalizes Hénon's ubiquitous 2d map and provides a local approximation for the dynamics of more general 4d maps. We show that the bounded dynamics of Moser's family is organized by a codimension-three bifurcation that creates four fixed points-a bifurcation analogous to a doubled, saddle-center-which we call a quadfurcation. In some sectors of parameter space a quadfurcation creates four fixed points from none, and in others it is the collision of a pair of fixed points that re-emerge as two or possibly four. In the simplest case the dynamics is similar to the cross product of a pair of Hénon maps, but more typically the stability of the created fixed points does not have this simple form. Up to two of the fixed points can be doubly elliptic and surrounded by bubbles of invariant two-tori; these dominate the set of bounded orbits. The quadfurcation can also create one or two complex-unstable (Krein) fixed points. Special cases of the quadfurcation correspond to a pair of weakly coupled Hénon maps near their saddle-center bifurcations.