The thing could not exist in real world; it is imaginary concept, expressed in a equations in a paper, that could run as a simulation on a PC. That could be interesting for people with some mathematical/physical/electrical background, and would require some knowledge in fields of linear algebra, EE, maybe circuit analysis..
To start, what is a quaternion? Well, in a simple words, its a number, that consist of 4 numbers: a "number" plus a imaginary 3D vector (that consists of the resting 3 numbers). There is a great video, interactively explaining "why and how"; with really great educational animations. What these 4D quaternions are able to represent - are a uniform scaling and rotations in 3D world.
As per electricity, wikipedia defines electric charge and current as well; Short video is also possible to look at, in order to get some clues. So in a real world, we have some wire, that transfers electrons, and we say that "charge flows", and the "rate of that flow" is a "current". So you see that this concept alredy became imaginary, but that's just a side note. "Amount of charge" and "rate of flow" are definetely some measurable quantities, that we represent as a real numbers. Then, on the papers, we've built an equation systems, that defines laws - Ohm's law and a Watt's law. The charge and current interactы with wires(resistors) and simple elements, like capacitors and inductors according to that laws:
Now lets imagine another world, lets call it "quorld", where exists some analog to charge, called "quarge". The "quarge" related to quorld's matter, "quatter" in a same way, that a charge relates to a matter in our world. Same laws are applied, with one exception. A quarge quantity is being measured by quaternion, instead of a real number. Thinking further on, there exists some qurrent - "a rate of flow of a quarge", and people queople in a quorld has invented their elements - quapacitors, that stores quarge, so on..
And here comes a fact, rendering above chart useless. A feature of quaternions is that multiplication of two quaternions is noncommutative;
I, multiplied by R is not equal to
R, multiplied by I
So, the order of multiplications does matters. It could be even helpful to define conjugate: a quaternion with inversed imaginary part.
So the conjugate of a product of two quaternions is the product of the conjugates in the reverse order:
(RI)* = I* R*
Also, there is another helpful property that they are associative:
(RI)I = R(II)
My ultimate goal is to write simulator, that can simulate quaternionic circuits, quircuits. And to check is there any mathematical contradictions, that proves non-functionality, or impossibility of existing such a system mathematically.
That involves a task of writing those 12 equations in a quaternionic form, with a correct multiplication order, as well as writing differential equations for a components: quapacitor, qinductor, qresistor(are they same component in that world?).
So if you have some math/physics/EE background and skills - you're welcome to assist me in those, that's primary goal of the question. I am a programmer, and for now I have web browser program with prototype, that is currently working with complex numbers. I would update with a working prototype, if question would be appreciated.
Also any thoughts on the topic are welcome, for example, how much dimensions such world could have? Is a mass "quass" also a quaternionic quantity in such a world? What are consequences of this, any thoughts, so on..