  In this paper we apply a homologous version of the Cauchy integral formula
for octonionic monogenic functions to introduce for this class of functions the
notion of multiplicity of zeroes and $a$-points in the sense of the topological
mapping degree. As a big novelty we also address the case of zeroes lying on
certain classes of compact zero varieties. This case has not even been studied
in the associative Clifford analysis setting so far.
  We also prove an argument principle for octonionic monogenic functions for
isolated zeroes and for non-isolated compact zero sets. In the isolated case we
can use this tool to prove a generalized octonionic Rouch\'e's theorem by a
homotopic argument. As an application we set up a generalized version of
Hurwitz theorem which is also a novelty even for the Clifford analysis case.
