  A formulation of quaternionic quantum mechanics ($\mathbb{H}$QM) is presented
in terms of a real Hilbert space. Using a physically motivated scalar product,
we prove the spectral theorem and obtain a novel quaternionic Fourier series.
After a brief discussion on unitary operators in this formalism, we conclude
that this quantum theory is indeed consistent, and can be a valuable tool in
the search for new physics.
