According to this post, it is not true that Cauchy completeness (every Cauchy sequence has a limit) and Dedekind completeness (every nonempty set that is bounded above has a supremum) are equivalent for ordered fields. The link in the post is sadly not working (I wish people kept their notes up). The user says,
Since the Archimedean hypothesis often goes almost without saying in calculus / analysis courses, many otherwise learned people are unaware that there are non-Archimedean sequentially complete fields. In fact there are rather a lot of them, and they can differ quite a lot in their behavior: e.g. some of them are first countable in the induced (order) topology, and some of them are not.
This is actually surprising to me, because I had a contrary false misconception. I think my misconception comes from the mantra that "$\mathbb{R}$ is the unique complete ordered field," which is misleadingly false unless we either refer to Dedekind completeness (which is usually termed as the least upperbound property) or we are presupposing the Archimedean property (which to be honest should be stated explicitly).
So I'm very curious, what are the examples of Cauchy complete ordered fields that are not isomorphic to $\mathbb{R}$? Can we make a list of these fields?