By now, there are lots of known results on real planar Hamiltonian systems but very few known results on complex planar Hamiltonian systems. The study of the dynamics of complex planar Hamiltonian systems has not only its own theoretical importance but also many practical applications. This work focuses on the global dynamics of complex planar Hamiltonian polynomial systems. Firstly, for general complex quadratic Hamiltonian systems of one degree of freedom, the authors give some sufficient conditions on the existence of family of invariant tori. Secondly, they complete the characterization of locally analytic linearizability of complex planar Hamiltonian systems with homogeneous nonlinearity of degrees either 2 or 3 at a nondegenerate singularity. Their global dynamics is investigated. Finally, for these classes of systems, they prove the existence of families of invariant tori, together with isochronous periodic orbits.