We study isochronous centres of plane polynomial Hamiltonian systems, and more generally, isochronous Morse critical points of complex polynomial Hamiltonian functions. Our first result is that if the Hamiltonian function H is a non-degenerate semi-weighted homogeneous polynomial, then it cannot have an isochronous Morse critical point, unless the associate Hamiltonian system is linear, that is to say H is of degree two. Our second result gives a topological obstruction for isochronicity. Namely, let be a continuous family of one-cycles contained in the complex level set , and vanishing at an isochronous Morse critical point of H, as . We prove that if H is a good polynomial with only simple isolated critical points and the level set contains a single critical point, then represents a zero homology cycle on the Riemann surface of the algebraic curve . We give several examples of `non-trivial' complex Hamiltonians with isochronous Morse critical points and explain how their study is related to the famous Jacobian conjecture.