Classical motion of complex 1D non-Hermitian Hamiltonian systems is investigated analytically to identify periodic, unbounded and chaotic trajectories. Expressions for the Lyapunov exponent for 1D complex Hamiltonians are derived. Complex potentials and V₂(x) = μx³ are studied in detail and their Lyapunov exponents are obtained analytically. It was found that when μ is complex all the trajectories of V₁ are chaotic with Lyapunov exponent |Im(μ)| and most of the trajectories of V₂ are periodic when μ is pure imaginary. But for other complex values of μ trajectories of V₂ are non-periodic and show infinite oscillations. Unbounded neighbouring trajectories of V₂ show power-law divergence rather than exponential divergence as in the case of V₁.