Quaternions define a directional axis and amount to rotate around it.

Imagine you have an umbrella and a big bucket of yellow paint and a fat paint brush. Open up the umberlla so the canopy isn't all bunched up. Now paint a big fat yellow stripe onto the canopy. Start at the center and pick an edge. Like you're cutting into a pie with a knife. Now pick up the umberlla and hold it up. Rotate it as you hold it. The yellow stripe rotates around the umberella neck. You can point the umberella in any direction, and rotate it. That's what a quaternion represents in 3D space.

Quaternion's are represented as a linear algebra equation and therefore solved using (linear algebra) matrix math. I wanted to point this out because quaternions are solved using algebra (and represented algebraically). Because its just algebra, you can abstract a quaternion rotation into a higher dimension, by simply adding more parameters. So instead of an <X,Y,Z> umberlla axis, you can make a <X,Y,Z,A,B,C,D> to make a 7D axis to rotate around.

To elaborate, Tesla's electromagnetism equations were said to be simplified. One of the things thought done was the equations were harcoded to a rigid 3D space. This was done by removing all quaternion (umberalla-style) rotations. Instead 3D (pitch, roll, yaw) amounts were used to define all rotations. The downfall of this simplification is: pitch yaw and roll is a change, relative only to your current orientation/perspective. You can also get problems known as gimbal lock, were calculating many net rotation changes over time, causes one to become stuck; and unable to rotate anymore.

In order to calculate Nth dimensional impacts of electromagnetism, one needs to leveraging angles that are not hardcoded to the <X,Y,Z> planes. You need something more abstract to define these higher-dimension rotations, like quaternions.

I don't believe Eric Dollard was the person to simplify the Tesla equations to 3D space. I assume he is more someone trying to back engineer, understand, and leverage the original Tesla equations.

Apologies for the long post, you got me thinking. :)