I think the paper below by Marek Danielewski is quite fundamental and important, so I thought I would try to "narrate" it and put it on youtube :-). For any details (in particular the math!), I would encourage you to read the paper - I focus mainly on the main concepts to hopefully get accross what this is all about.
Foundations of the Quaternion Quantum Mechanics (publication):https://www.mdpi.com/1099-4300/22/12/1424#
We show that quaternion quantum mechanics has well-founded mathematical roots and can be derived from the model of the elastic continuum by French mathematician Augustin Cauchy, i.e., it can be regarded as representing the physical reality of elastic continuum. Starting from the Cauchy theory (classical balance equations for isotropic Cauchy-elastic material) and using the Hamilton quaternion algebra, we present a rigorous derivation of the quaternion form of the non- and relativistic wave equations. The family of the wave equations and the Poisson equation are a straightforward consequence of the quaternion representation of the Cauchy model of the elastic continuum. This is the most general kind of quantum mechanics possessing the same kind of calculus of assertions as conventional quantum mechanics. The problem of the Schrödinger equation, where imaginary ‘i’ should emerge, is solved. This interpretation is a serious attempt to describe the ontology of quantum mechanics, and demonstrates that, besides Bohmian mechanics, the complete ontological interpretations of quantum theory exists. The model can be generalized and falsified. To ensure this theory to be true, we specified problems, allowing exposing its falsity.
Interactive videos to understand quaternions:https://eater.net/quaternions
More links and references:
Foundations of the Quaternion Quantum Mechanics (publication): https://www.mdpi.com/1099-4300/22/12/1424# https://www.researchgate.net/profile/Danielewski_Marek/research
Slides for “Foundations of Quantum Mechanics and Gravity”: https://www.dropbox.com/s/j3de24nnvjc9g07/Time_2020_2a.ppt
M. Danielewski, “The Planck-Kleinert Crystal”, Z. Naturforsch. 62a, 564-568 (2007).
H. Kleinert & J. Zaanen, „Nematic world crystal model of gravity explaining absence of torsion in spacetime”, Physics Letters A 324 (2004) 361–365.http://www.classicalmatter.org/IntroWaveMech.pdfhttps://eater.net/quaternionshttps://www.wired.com/story/meet-the-four-dimensional-numbers-that-led-to-modern-algebra/
QM Seminar: http://th.if.uj.edu.pl/~dudaj/QMFNoThttps://en.wikipedia.org/wiki/Gimbal_lock#:~:text=Gimbal%20lock%20is%20the%20loss,misleading%3A%20no%20gimbal%20is%20restrained.https://demonstrations.wolfram.com/RotatingACubeUsingQuaternions/https://quaternions.online/http://www.askingwhy.org/blog/
(a summary of some of my questions and links)
Table of Contents:
00:00 - Quaternion Quantum Mechanics
00:35 - Overview
05:14 - Phonons
09:41 - Minkowski space-time
10:07 - Planck Kleinert Crystal: space-Density
13:21 - Optical Black Holes
14:28 - Properties of the Planck Kleinert Crystal
19:27 - Motivation 1 for Quaternions: gimbal lock
20:30 - Quaternions
27:13 - Deformation Fields
28:58 - Cauchy
34:36 - Cauchy and Helmholtz
39:21 - Constants of the Planck Kleinert Crystal
43:50 - Energy of deformation field
46:44 - Schrödinger Equation
48:26 - What is spin?
50:23 - What about special relativity?
56:20 - Links