Real Monge-Ampère operators and (almost) complex structures

We observe some interesting geometric structures which are naturally linked with the geometric approach to Monge-Ampère operators developed by Lychagin in late 70th.
Among them are: (almost) complex, (almost) product, generalized complex, hyperkahler, hypersymplectic and many other geometric structures.
I hope (if I have time) to show few interesting examples of its applications.