In this paper we give a numerical description of the neighbourhood of a fixed point of a symplectic map undergoing a transition from linear stability to complex instability, i.e., the so called Hamiltonian-Hopf bifurcation. We have considered both the direct and inverse cases. The study is based on the numerical computation of the Lyapunov families of invariant curves near the fixed point. We show how these families, jointly with their invariant manifolds and the invariant manifolds of the fixed point organise the phase space around the bifurcation.