  In this article, we completely characterize the complex symmetry, cyclicity
and hypercyclicity of composition operators $C_\phi f=f\circ\phi$ induced by
affine self-maps $\phi$ of the right half-plane $\mathbb{C}_+$ on the
Hardy-Hilbert space $H^2(\mathbb{C}_+)$. We also provide new proofs for the
normal, self-adjoint and unitary cases and for an adjoint formula discovered by
Gallardo-Guti\'{e}rrez and Montes-Rodr\'{i}gues.
