  Hurwitz (1887) defined a continued fraction algorithm for complex numbers
which is better behaved in many respects than a more "natural" extension of the
classical continued fraction algorithm to the complex plane would be. Although
the Hurwitz complex continued fraction algorithm is not "reducible" to another
complex continued fraction algorithm, over the reals the story is different. In
this note we make clear the relation between the restriction of Hurwitz's
algorithm to the real numbers and the classical continued fraction algorithm.
As an application we reprove the main result of Choudhuri and Dani (2015).
