This paper presents an application of hypercomplex algebras combined with dictionary learning for sparse representation of multichannel images. Two main representatives of hypercomplex algebras, Clifford algebras and algebras generated by the Cayley-Dickson procedure are considered. Related works reported quaternion methods (for color images) and octonion methods, which are applicable to images with up to 7 channels. We show that the current constructions cannot be generalized to dimensions above eight.