  We develop a formalism for describing the most general notion of tree-level
scattering amplitudes in 4d conformal higher spin theory. As conformal higher
spin fields obey higher-derivative equations of motion, there are many distinct
on-shell external states which may contribute to their scattering, some of
which grow polynomially with time, leading to ill-defined amplitudes. We
characterize the set of admissible scattering states which produce finite tree
amplitudes, noting that there are more such states than just standard massless
higher spins obeying two-derivative equations of motion. We use conformal
gravity as a prime example, where the set of scattering states includes the
usual Einstein graviton and a `ghost' massless spin 1 particle. An extension of
the usual spinor helicity formalism allows us to encode these scattering states
efficiently in terms of `twistor-spinors'. This leads to compact momentum space
expressions for all finite tree-level 3-point amplitudes of conformal higher
spin theory. While some of these 3-point amplitudes vanish (including all those
with only standard two-derivative higher spin external states), there are many
others which are non-vanishing. We also comment on the generalization to
scattering of conformal higher spins in AdS$_4$.
