The mathematics of evolving fields with two complex components is known already in Jones calculus.
This blog entry delivers an other example of an elliptic complex which can be used in discrete Atiyah-Singer or Atiyah-Bott type setups as examples. We had seen that when deforming an elliptic complex with an integrable Lax deformation, we get complex elliptic complexes. We had wondered in that blog entry whether a complex can lead to quaternion-valued fields. The discussion … ….