  Separatrices divide the phase space of some holomorphic dynamical systems
into separate basins of attraction or 'stability regions' for distinct fixed
points. 'Bundling' (high density) and mutual 'repulsion' of trajectories are
often observed at separatrices in phase portraits, but their global
mathematical characterisation is a difficult problem. For 1-D complex
polynomial dynamical systems we prove the existence of a separatrix for each
critical point at infinity via transformation to the Poincar\'e sphere. We show
that introduction of complex time allows a significantly extended view with the
study of corresponding Riemann surface solutions, their topology, geometry and
their bifurcations/ramifications related to separatrices. We build a bridge to
the Riemann $\xi$-function and present a polynomial approximation of its Newton
flow solution manifold with precision depending on the polynomial degree.
