  In this paper we revisit the concept of conformality in the sense of Gauss in
the context of octonions and Clifford algebras. We extend a characterization of
conformality in terms of a system of partial differential equations and
differential forms using special orthonormal sets of continuous functions that
have been used before in the particular quaternionic setting. The aim is to
describe to which higher dimensional algebras this characterization can exactly
be extended and under which circumstances. It turns out to be crucial that this
characterization requires a domain of definition that lies in a subalgebra that
has the norm composition property and that is either associative (Clifford
algebra case) or at least alternative (octonionic case). The orthonormal frames
are elements of the spin group Spin(n+1). We round off by relating the nature
of the orthonormal frames to the associated M\"obius transformation which are
related to SO(9,1) in the octonionic case and to the Ahlfors-Vahlen group in
the case of a Clifford algebra.
