When definining the completion of a field $k$ by a norm one typically uses Cauchy sequences. More specifically the completion of $k$ is defined as the set of equivalence classes of Cauchy sequences in $k$. It seems natural that one would want Cauchy sequences to be convergent in order to say that the field "does not have holes". However I have never seen a justification that would make Cauchy sequences a canonical choice of type of sequence that should be used in the definition of completion. Perhaps there are other equally valid choices to define a completion and they would yield a non-isomorphic field?
Are Cauchy sequences indeed a canonical choice for a definition of the completion? Is there a formulation of the completion by a norm as a universal object in some category or any way to fit Cauchy sequences into a natural setting?