If you start with NPA it is identical as PA. You can apply all the same logic and all the first theorems. The problem with multiplication is not that the proofs fall apart, it is that multiplication does not mean what you want it to mean.
When you go to the complex numbers, you have a similar situation. The only thing that distinguishes i from -i is what you want the principal branch of sqrt to be. Otherwise nothing would change if you flip them.
> The problem with multiplication is not that the proofs fall apart, it is that multiplication does not mean what you want it to mean.
Well, yeah, that's kind of the point: the positiveness of the naturals is not just a convention, it is actually a logical consequence of something in the axioms. Specifically, it is a logical consequence of the definition of multiplication. I find that interesting.
That ties to one of the standard axioms of the real numbers in classical mathematics.
There exists a set P which is closed under addition and multiplication, such that for every real number x, one of these three things is true: x is in P, -x is in P, x = 0.
P is, of course, the positive numbers. This axiom makes the reals into an ordered field. Add in completeness, and the reals are unique (up to choice of set theory).
When you go to the complex numbers, you have a similar situation. The only thing that distinguishes i from -i is what you want the principal branch of sqrt to be. Otherwise nothing would change if you flip them.