A field (as an algebraic structure) equipped with a metric is complete if the operations are continuous with respect to the induced topology and it is a complete metric space.
The most important case is the case of valued fields.
Given a multiplicative discrete valuation on a field , there is a topology on induced by the metrics induced by the valuation. A complete valued field is a field complete with respect to the valuation metric.
One of Ostrowski's theorems says that for a field complete with respect to an absolute value either the absolute value is archimedean, in which case is either the field of real numbers or of complex numbers, or the absolute value is non-archimedean.