  We prove that for every discrete-time linear switching system in two complex
variables and with finitely many switching states, either the system is
Lyapunov stable or there exists a trajectory which escapes to infinity with at
least linear speed. We also give a checkable algebraic criterion to distinguish
these two cases. This dichotomy was previously known to hold for systems in two
real variables, but is known to be false in higher dimensions and for systems
with infinitely many switching states.
