Summary: We construct a Hamiltonian suspension for a given symplectomorphism which is the perturbation of a Poincaré map. This is especially useful for the conversion of perturbative results between symplectomorphisms and Hamiltonian flows in any dimension \(2d\). As an application, using known properties of area-preserving maps, we prove that for any Hamiltonian defined on a symplectic 4-manifold \(M\) and any point \(p \in M\), there exists a \(C^{2}\)-close Hamiltonian whose regular energy surface through \(p\) is either Anosov or contains a homoclinic tangency.