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Model completion is a term employed when Existential Closure is successful. The formation of the
Complex Numbers, and the move from affine to projective geometry, are successes of this kind.
The theory of existential closure gives a theoretical basis of Hilbert's 
``method of ideal elements.''
References
Manders, K. L. ``Interpretations and the Model Theory of the Classical Geometries.'' In Models and Sets.
Berlin: Springer-Verlag, pp. 297-330, 1984.
Manders, K. L. ``Domain Extension and the Philosophy of Mathematics.'' J. Philos. 86, 553-562, 1989.