You argued (without much detail, but I'll buy it) for completeness, but then snuck in ordered-ness and Archimedean-ness. As a p-adic analyst, I object to taking these latter two characteristics for granted as necessary for mathematics.
(Also, one has to be careful about the meaning of 'complete' in the uniqueness statement; i.e., it must be understood to mean complete as an ordered field (satisfying the least-upper-bound property), not just complete as a uniform space (having every Cauchy sequence converge). See Paul Sally's "Tools of the trade" (http://www.amazon.com/Tools-Trade-Paul-J-Sally/dp/0821846345) for some discussion of this.)
(Also, one has to be careful about the meaning of 'complete' in the uniqueness statement; i.e., it must be understood to mean complete as an ordered field (satisfying the least-upper-bound property), not just complete as a uniform space (having every Cauchy sequence converge). See Paul Sally's "Tools of the trade" (http://www.amazon.com/Tools-Trade-Paul-J-Sally/dp/0821846345) for some discussion of this.)