" tommy quaternion "

[Catullus]

Tommy, to be honest I just think that given like that, it an seems arbitrary set of conditions.
I don't quite get the red line that is connecting all those properties together and connecting the various number systems nor what you are looking for. I believe this is part of some quest you are undertaking, but I believe you have not made the goal explicit. What you are going for Tommy?

Sure it's a field in its own interesting and it "deserves more attention" but in giving these definition your going random or following some kind of map/logic?

I ask because there are infinite kinds of algebraic structures, and since it is abstract algebra, if you have not a chart is easy to get lost into meaningless abstraction (category theory was born for this reason, in order to do not get lost into Hilbert/Bourbaki's kind of structural abstractness, in that it gives you conceptual tool for mapping and exploring the territory).

Happy 250th post! 
You are a Long Time Fellow now!