Using as information-quantifiers the entropy and the statistical complexity, we analyze the rich, complex dynamics of a special non linear Hamiltonian H. H describes the interaction between a quantum system and a classical one. The concomitant system exhibits periodicity, quasi-periodicity, not-boundedness, and chaotic regimes. The chaotic phenomenon, together with complex dynamics, arise in the vicinity of an unstable case, that of the purely quantum system.2) Complexity of a matter-field Hamiltonian in the vicinity of aquantum instability.