  We find a compactification of the $\mathrm{SO}_{0}(2,3)$-Hitchin component by
studying the degeneration of the induced metric on the unique equivariant
maximal surface in the 4-dimensional pseudo-hyperbolic space
$\mathbb{H}^{2,2}$. In the process, we establish the closure in the space of
projectivized geodesic currents of the space of flat metrics induced by
holomorphic quartic differentials on a Riemann surface. As an application, we
describe the behavior of the entropy of Hitchin representations along rays of
quartic differentials.
