Why do we not study the behavior of a system of ODEs around complex equilibrium points? How does their existence influence the flow?
I have studied the stability and bifurcation analysis for small systems. Perhaps the "Ghosts" in bifurcation diagrams from Strogatz's book may give some context.
Practically speaking, we don't care about the existence of complex equilibrium points because all the realizable models have real signals or states which means that the states can not reach these equilibrium points if they are not real. There are no any kind of complex signals despite of the fact that we represent some concepts in electrical analysis using imaginary number as the reactive power but this still be a representation of a concept that is not a signal or a state of the model.
Also there is a big difference between the existence of the complex eigenvalues of the state matrix and the equilibrium because the eigenvalues are used to analyse the behavior of the dynamical system and they are not values of the states in the system. So, the usage of complex analysis is just to analyze systems.
On the other hand; from the definition of the equilibrium point: it is the point that if the states at that point stay forever, so the states must be complex and this is impossible in practice.