  We perform an exact renormalization-group analysis of one-dimensional 4-state
clock models with complex interactions. Our aim is to provide a simple explicit
illustration of the behavior of the renormalization-group flow in a system
exhibiting a rich phase diagram. In particular we study the flow in the
vicinity of phase transitions with a first-order character, a matter that has
been controversial for years. We observe that the flow is continuous and
single-valued, even on the phase transition surface, provided that the
renormalized Hamiltonian exist. The characteristics of such a flow are in
agreement with the Nienhuis-Nauenberg standard scenario, and in disagreement
with the ``discontinuity scenario'' proposed by some authors and recently
disproved by van Enter, Fern\'andez and Sokal for a large class of models (with
real interactions). However, there are some points in the space of interactions
for which a renormalized Hamiltonian cannot be defined. This pathological
behavior is similar, and in some sense complementary, to the one pointed out by
Griffiths, Pearce and Israel for Ising models. We explicitly see that if the
transformation is truncated so as to preserve a Hamiltonian description, the
resulting flow becomes discontinuous and multivalued at some of these points.
This suggests a possible explanation for the numerical results that motivated
the ``discontinuity scenario''.
