Let \(\mathbb{O}\) denote the real algebra of octonions, or Cayley numbers, \[x=x_0+\sum_{t=1}^7x_te_t,\] with \(x_t \in \mathbb{R}\) for all \(t \in \{0,\ldots,7\}\). The article studies {left (or right) octonionic monogenic} functions: namely, solutions of \(\mathcal{D}f=0\) (or \(f\mathcal{D}=0\)), where \[\mathcal{D}:=\frac{\partial}{\partial x_0}+\sum_{t=1}^7e_t\frac{\partial}{\partial x_t}\] is the octonionic Cauchy-Riemann operator. This notion, attributed to [K. Imaeda and M. Imaeda, Appl. Math. Comput. 115, No. 2–3, 77–88 (2000; Zbl 1032.17003); X. Li and L. Peng, Approximation Theory Appl. 16, No. 2, 28–36 (2000; Zbl 0976.31008)], is distinct from the notion of octonionic regularity of G. Gentili and D. C. Struppa [Rocky Mt. J. Math. 40, No. 1, 225–241 (2010; Zbl 1193.30070)].
The author recalls several properties of octonionic monogenic functions known in literature and provides a proof of their Identity Principle. For each octonionic monogenic function \(f\) and each value \(a\) of \(f\), he defines and studies the {order} of \(f\) at an isolated point of the preimage \(f^{-1}(a)\). For the case when all points considered are isolated, he proves an Argument Principle and a version of Rouché’s Theorem (whence a version of Hurwitz’s Theorem). The final part of the article sets the framework to address the case of non-isolated points of the preimage \(f^{-1}(0)\) and generalizes the octonionic Argument Principle within this framework. The study of other properties within the same framework is left as an open problem.