N=8 Supersymmetric Quaternionic Mechanics

We construct N=8 supersymmetric mechanics with four bosonic end eight fermionic physical degrees of freedom. Starting from the most general N=4 superspace action in harmonic superspace for the ({\bf 4,8,4}) supermultiplet we find conditions which make it N=8 invariant. We introduce in the action Fayet-Iliopoulos terms which give rise to potential terms. We present the action in components and give explicit expressions for the Hamiltonian and Poisson brackets. Finally we discuss the possibility of N=9 supersymmetric mechanics.