Once you replace scalars with 2 vectors you aren't basing it on real numbers, you're basing it on operations on 2 vectors. Of course you can put real numbers a

Once you replace scalars with 2 vectors you aren't basing it on real numbers, you're basing it on operations on 2 vectors. Of course you can put real numbers at the base of nearly anything, but it the theory operates at a fundamental level on items that are more complex then real numbers, then it's not based on real numbers. You might as well argue it's based on surreal numbers or Dedekind cuts at that point. Somewhere Feynman has a quote that qm was the first theory that required complex numbers. Finding and understand that quote should explain this better.