Other 2D extensions of the reals

Question

The complex numbers are one instance of an extension of the reals. But are there others? For example, with complex numbers, $i^2=-1$, but there are other ways of defining $i$ which also result in an abelian group. Five other instances are

1. $i^2=-1-i$
2. $i^2=1$
3. $i^2=-1+i$
4. $i^2=1-i$
5. $i^2=1+i$

There are also some extensions which are abelian, except for invertablity. Some of these are

1. $i^2=0$
2. $i^2=i$
3. $i^2=-i$

So, are these valid extensions of the reals? If they are, what makes $i^2=-1$ special compared to the rest of them?