Ed,
Many thanks for reading and commenting.
I think you are correct regarding EM and gravity. The Kaluza-Klein Equation is a 5-D model that combines EM with gravity. It was abandoned because AE believed that the implied scalar field was not compatible with GR.
I also think you are correct regarding quaternions and Minkowski space-time. The scalar term in a quaternion can be used to relate the dot product of a vector and the change in that vector to the length of the vector.
Having stated that, I would like to point out that the relativistic energy equation can be produced by setting Q and Q* in my Equation 5.2 as follows:
Q = m_0*c^2 p*c
Q* = m_0*c^2 - p*c
Here, m_0 and c are scalars and p is a vector.
This implies a velocity quaternion as follows:
V = c v
Here, c is a scalar and v is a vector.
I think this velocity quaternion is at the heart of the Hertz Equation Galilean Transform that you demonstrated.
If I then integrate that velocity quaternion with respect to scalar t, the following results:
Vt = X = ct vt
If this is an indefinite integral then there could be a constant in there. If it is a definite integral, then this will be the difference between final and initial conditions.
So, this is a quaternion that represents space-time, but not Minkowski space-time. Whether or not this is actually Physics is another question.
I have given some thought to how to fit all of this together. The main stumbling block that I see is that when two bi-quaternions are multiplied together, the result has four terms (AC, BC, BD, and AD). Each of these must represent something that is physically real and measureable.
I have begun to study Dr. Hestene's work. It will take me several years to build a satisfactory level of knowledge.
Best Regards and Many Thanks,
Gary Simpson