We can find a set that is not $\mathbb{R}$ which satisfies the field axioms and is a totally ordered set, however we cannot find one that is Dedekind complete. Up to isomorphism $\mathbb{R}$ is the only Dedekind complete ordered field, and this is actually its defining property in a significant sense.
The study of real-closed fields may be of some interest to you; it is the first widely recognized generalization of the study of $\mathbb{R}$, looking at totally ordered fields with the same first order theory as the real numbers. This means that any statement $\phi$ which is true about elements of the real numbers must also be true for the elements of this other field, $\mathbb{F}$.
Completeness is a topological property of $\mathbb{R}$ and consequently part of its second order theory, the theory of subsets of $\mathbb{R}$. There are many real-closed fields which are not $\mathbb{R}$ -- the smallest among them is $\overline{\mathbb{Q}}$, the algebraic closure of the rational numbers, and the largest real-closed field is the Surreal numbers $N_0$. We also have that $\overline{\mathbb{Q}}\subset\mathbb{R}\subset N_0$, in the sense that there is a strict subfield of $N_0$ which is isomorphic to $\mathbb{R}$ and a strict subfield of $\mathbb{R}$ which is isomorphic to $\overline{\mathbb{Q}}$.
If you would like to see more examples I construct a proper class of non-isomorphic real-closed fields in this paper https://arxiv.org/abs/1706.08908, the smallest one appearing in here being $\mathbb{R}$ and the largest being $N_0$ -- this part of the construction begins on page $53$.
