Quaternionic Computing

We introduce a model of computation based on quaternions, which is inspired on the quantum computing model. Pure states are vectors of a suitable linear space over the quaternions. Every other aspect of the theory is the same as in quantum computing: superposition and linearity of the state space, unitarity of the transformations, and projective measurements. We then show that this model is no more powerful than quantum computing. More concretely we show, that any quaternionic computation using n quaterbits, can be simulated with n+1 qubits, and this with little or no overhead in circuit size. Furthermore, by using the same technique, we provide a new proof that a similar model based on real amplitudes is not weaker than quantum computing, since any quantum computation using n qubits can be simulated with n+1 rebits, and in this with no circuit size overhead.