I hate to “pull rank” but I have a PhD in mathematics so I’m personally well aware of all the nuances. Again my point is historical/pedagogical. Why should “pair numbers” or “double numbers” be the answer as the OP suggested? It’s not straightforward without getting into conformal mappings.
And why is two dimensions enough for third and higher degree equations? Were there a simpler geometric connection, you’d probably have a nicer proof of the Jordan curve theorem… but you don’t.
Not sure what you mean by pair of pairs… the OP said that calling imaginary numbers “double numbers” would clear up a lot of issues but that is not clear at all to me.
And why is two dimensions enough for third and higher degree equations? Were there a simpler geometric connection, you’d probably have a nicer proof of the Jordan curve theorem… but you don’t.
Not sure what you mean by pair of pairs… the OP said that calling imaginary numbers “double numbers” would clear up a lot of issues but that is not clear at all to me.