If we consider the extra dimension(s) on which the amplitude of the wave function given to the Schrodinger Equation, the wave function instead defines a topology-like geometric object. If the topology can be evaluated over time by some alternative mathematical construct, that alternative mathematical construct may form the basis for a more powerful (in the sense of describing a wider range of potential phenomena) physics, because it should be constructable in such a way as to not possess the limitations of the Schrodinger Equation that the function returns a value for the entire dimensional space under consideration.
The amplitude of quantum waves are geometrically limited in a way that the topology shouldn't be; quantum waves have an extent from 0 to the amplitude, whereas a more general topology should permit discontinuous extents. The amplitude of the probability would be equivalent to a thickness, or measure, in the topology; the exact position of the topology relative to the dimension of amplitude could vary (this variance is why I describe this as a topology). These values could potentially matter for the generalized version of the Schrodinger Equation, which would describe the special case where these values don't matter, where the measure is continuous, and where the topology is defined for all of the non-amplitude dimensions.
I suspect there is a class of such generalizations, which make distinct predictions.
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