It is a well-known fact that Euclidean geometry and the arithmetic of real numbers are both decidable and complete theories. Here are my questions:
- Do the non-Euclidean geometries share the same properties?
- Does the arithmetic of complex numbers, quaternions, etc share the same properties?
I think the answer to 1 is 'yes' because the non-Euclidean geometries are interpretable in Euclidean geometry. I also think that the arithmetic of complex numbers, quaternions, etc. share these properties as well because it seems impossible to define natural numbers in these number systems, so the Gödel's incompleteness theorems don't go through.
Am I right?
The theory of algebraically closed fields of characteristic $0$ is complete and decidable. So yes, one can say that the arithmetic of complex numbers is complete and decidable.
For quaternions, one can interpret a sentence $\varphi$ involving say addition, multiplication, and the "special" quaternions as a sentence about the reals. So decidability is inherited from the decidability of the theory of real-closed fields.
The first-order theory of hyperbolic geometry is decidable. For a link please see this.