This is solid a description of the family of approaches being used, circa 1993. See the text for details. Prima Facie Questions in Quantum Gravity Isham is concerned with the form of these approaches. My framework was stated explicitly above, but it literally does not appear on the radar screen of this discussion of quantum gravity. This statement makes a hidden assumption, that quaternions do not belong on a list that includes real and complex numbers.
Quaternions have the same continuum properties as the real and complex numbers.
The important distinction is that quaternions do not commute. This property is shared by quantum mechanics so it should not banish quaternions from the list. The omission reflects the history of work in the field, not the logic of the mathematical statement. General relativity may force non-linearity into quantum theory, which require a change in the formalism. It is easy to write non-linear quaternion functions.