People can already count since several millennia. Arithmetic is improved over time. Usually we count by using integers. If we want to get more accurate results, then we better use rational numbers. Sometimes we also use numbers that are not expressible in a fraction, such as the number π (pi) and the square root of two . These are real numbers. The square root of a negative number delivers more problems. For this task it is necessary that we move to a two-dimensional number system. The better calculators under us can also handle this problem fluently. The so-called complex numbers are discovered in the sixteenth century by the Italian mathematician Gerolamo Cardano and are now used for all kinds of applications. Especially wave movements and other oscillations are easily described by complex numbers. It took until half of the nineteenth century before William Rowan Hamilton discovered the successor number collection. These numbers are not three but four-dimensional and are called quaternions. Quickly the eight dimensional octonions followed. This indicates that the dimension of the number systems increase by a factor of two. As the dimension increases, the computational capabilities decrease. For example, the quaternions can no longer be multiplied in random order.
A quaternion can be seen as the combination of a real number and a three dimensional vector. With their 1 +3D structure quaternions seem well suited for application in physics. Shortly after the discovery of quaternions many physicists have tried this. Remarkably,this failed miserably. The reason is the less convenient quaternionic product rule. In many cases it much easier to apply complex numbers or vector analysis than quaternions. It quickly made the quaternions redundant. This is intensified by the fact that the special theory of relativity of Einstein claims a role for the space-time combination where quaternions do not fit in. The also 1 + 3D spacetime has a so-called Minkowski signature, whereas quaternions have a Euclidean signature. That meant the actual death blow for the application of quaternions in physics. That this was in fact unjustified recognized only few scientists.
Quaternions prove very suitable as eigenvalues of operators that deliver these values as observable quantities. This cannot be said for spacetime. Thus spacetime is not a genuine observable. So, there must be more to it. However, this only becomes apparent when we dare to let quaternions play a greater role. Quaternions and especially functions with quaternionic values possess some unexpected properties that sofar have remained virtually unexposed.
Most physicists use complex functions in order to represent the state of small particles with it. In essence, these state functions are probability amplitude distributions. By using complex state functions, these functions can easily show the wave behavior of these particles. This is why the corresponding part of physics is called wave mechanics. However, it is equally well possible to represent the state of the particle by using a quaternionic function. This pure mathematical measure causes that what happens is no longer seen as a wave problem, but rather as a fluid problem. With respect to physical reality, this measure changes nothing. The only thing that differs is that the mathematical toolkit now provides a very different view about what happens. For most physicists this last method of viewing is utterly strange.
The unexpected move is due to the fact that quaternionic probability amplitude distributions can be considered to be a combination of a charge density distribution and a current density distribution or can be viewed as a combination of a scalar potential and a vector potential. The result is that when a switch from a complex state function is made to a quaternionic state function, any linear equation of motion is converted into a balance equation.
Fluid mechanics conventionally belongs to another part of physics than wave mechanics. However, the outlined transition lets wave mechanics convert in a kind of fluid dynamics. This is nothing short of a revolution. Instead of oscillations we see high-and low-pressure areas and currents. That is not to say that the oscillations are not there anymore. The only thing that happens is that with the new tools we perceive less well the oscillations, but we get a much better look at the currents and the high-and low-pressure areas.
A fact that is still not entirely addressed is formed by the sign choices that quaternionic functions offer. Quaternions offer four mutually independent sign choices. The first is called the conjugation. It changes the sign of the imaginary part. The three others are reflections. They change the sign in one of the three imaginary directions. Together, these sign choices provide eight different sign variants. Separately, the conjugation and the reflections cause a transition of the handedness of the quaternionic product. These properties show remarkable effects. This is reflected in the coupling of state functions as is described by equations of motion such as the Dirac and Majorana equations. This coupling causes a local compression of the space that surrounds the considered particles. It can be explained by fluid dynamics.
The conclusion may be drawn that ignoring quaternions and quaternionic functions was not justified. Correct use of these ingredients gives a sea of new opportunities for fundamental physics.
The new look is applied in http://www.crypts-of-physics.eu/Quaternionic_Quantum_Physics.pdf