  Treating the two-dimensional Minkowski space as a Wick rotated version of the
complex plane, we characterize the causal automorphisms in two-dimensional
Minkowski space as the M\"{a}rzke-Wheeler maps of a certain class of observers.
We also characterize the differentiable causal automorphisms of this space as
the Minkowski conformal maps whose restriction to the time axis belongs to the
class of observers mentioned above. We answer a recently raised question about
whether causal automorphisms are characterized by their wave equation. As
another application of the theory, we give a proper time formula for
accelerated observers which solves the twin paradox in two-dimensional
Minkowski spacetime.
