  We determine the exact complex-temperature (CT) phase diagram of the 1D $Z_6$
clock model. This is of interest because it is the first exactly solved system
with a CT phase boundary exhibiting a finite-$K$ intersection point where an
odd number of curves (namely, three) meet, and yields a deeper insight into
this phenomenon. Such intersection points occur in the 3D spin 1/2 Ising model
and appear to occur in the 2D spin 1 Ising model. Further, extending our
earlier work on the higher-spin Ising model, we point out an intriguing
connection between the CT phase diagrams for the 1D and 2D $Z_6$ clock models.
