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We use a large set N-body/hydrodynamical simulations to study the physical
properties of the merging cluster El Gordo. We found that the observed X-ray
structures, along with other data, can be fairly matched by simulations with
collision velocities 2,000 kms <= V <= 2,500 kms and impact parameters 600 kpc
<= P <= 800 kpc. The mass of the primary is constrained to be between 10^{15}
M_sun and ~ 1.6 10^{15} M_sun, in accord with recent lensing-based mass
measurements. Moreover, a returning, post-apocenter, scenario is not supported
by our head-on simulations. We also consider merger models that incorporate
dark matter self-interactions. The simulation results show that the observed
spatial offsets between the different mass components are well reproduced in
self-interacting dark matter models with an elastic cross-section in the range
\sigma_DM/m_X ~ 4 -5 cm^2/gr. In addition, the mean relative line-of-sight
radial velocity between the two brightest cluster galaxies is found to be of
the order of several hundreds of km/s. We argue that these findings provide an
unambiguous signature of a dark matter behavior that exhibits collisional
properties in a very energetic high-redshift cluster collision. The range of
allowed values we found for sigma_DM/m_X is however inconsistent with present
upper limits. To resolve this tension we suggest the possibility that the
self-interacting dark matter model used here should be considered as only a low
order approximation, and that the underlying physical processes that describe
the interaction of dark matter in major cluster mergers are more complex than
can be adequately represented by the commonly assumed approach based on
scattering of dark matter particles.