Interesting, but since there's no reason to multiply two money values together (what use is a square dollar?), it doesn't really do anything. It ends up trivially degenerating into two currencies with some exchange rate between them.
I already consider food to have costs in two currencies: one in dollars and one in calories. There is no conversion between them; I must meet both budgets.
What on Earth are you talking about "there's no conversion between them"?!? You can't trade dollars for food? You can't trade food for dollars? What are dollars good for if there's no conversion between them and food?
The lack of an abstract market selling and buying abstract "calories" at some price doesn't mean there isn't a conversion between them, just that it isn't clean and has a lot of other moving parts, just like everything else in the real world.
I don't mean that I can't buy food. I mean that I can't trade between dollars and calories. For example, a scoop of premium ice cream costs me $1.25 and 250 calories. The currencies are independant and inexchangeable; if there was an exchange rate between them, I could get that same food for $5.00 and 50 calories.
Any food I contemplate has a cost somewhere in dollar-calorie space, so the currency with which I buy food can be accurately modeled as a two dimensional vector space (or perhaps as the complex numbers, though I guess I don't really use the field properties...).
If you also introduce imaginary time (as in some models of the big bang), then you can find funny definitions for what it means to possess a certain amount of money for a certain amount of time shortly after the big bang.
Perhaps something for interdisciplinary theoretical economist-physicists.
