  We discover multi-Hamiltonian structure of complex Monge-Ampere equation
(CMA) set in a real first-order two-component form. Therefore, by Magri's
theorem this is a completely integrable system in four real dimensions. We
start with Lagrangian and Hamiltonian densities and obtain a symplectic form
and the Hamiltonian operator that determines the Dirac bracket. We have
calculated all point symmetries of two-component CMA system and Hamiltonians of
the symmetry flows. We have found two new real recursion operators for
symmetries which commute with the operator of a symmetry condition on solutions
of the CMA system. These operators form two Lax pairs for the two-component
system. The recursion operators, being applied to the first Hamiltonian
operator, generate infinitely many real Hamiltonian structures. We show how to
construct an infinite hierarchy of higher commuting flows together with the
corresponding infinite chain of their Hamiltonians.
