In this article we study function theory in the 16-dimensional space of complexified octonions \(\mathbb {O}_{\mathbb {C}}=\mathbb {C}\otimes \mathbb {O}\) . We define the complexified octonionic Cauchy–Riemann operator \( D=\sum _{j=0}^7 e_j\partial _{z_j} \) where \(\partial _{z_j}{:=}\partial _{x_j}+i\partial _{y_j}\) for \(j=0,1,...,7\) . This operator together with its octonionic conjugate factorize the ultrahyperbolic operator \( \mathcal {L}=D\overline{D}=\overline{D}D=\sum _{j=0}^7 \partial _{z_j}^2. \) In real coordinates we obtain a Lichnerowicz-Weitzenböck type formula of the form \( \mathcal {L} = \Delta _x - \Delta _y + 2i \langle \partial _x, \partial _y \rangle \) One main goal of this paper consists in investigating the fundamental solution and polynomial solutions of the operator \(\mathcal {L}\) . The set of polynomial solutions is completely described by proposing explicit basis constructions and dimension formulae. Another main goal that is to carefully figure out in detail how complexified octonion analysis precisely differs from classical real octonion analysis in 8 dimensions. We explain some really substantial differences. In particular, the component functions are not Euclidean harmonic but ultrahyperbolic harmonic exhibiting a very different regularity behavior. Furthermore, the classical Fischer-decomposition is not anymore true in this context.
