Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)A nonassociative Groenwold-Moyal (GM) plane is constructed using quaternion-valued function algebras. The symmetrized multiparticle states, the scalar product, the annihilation/creation algebra and the formulation in terms of a Hopf algebra are also developed. Nonassociative quantum algebras in terms of position and momentum operators are given as the simplest examples of a framework whose applications may involve string theory and nonlinear quantum field theory.2835Departamento de Fisica Matematica of the Universidade de Sao PuloCoordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)