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We propose to apply the idea of analytical continuation in the complex domainto the problem of geodesic completeness. We shall analyse rather in detail thecases of analytical warped products of real lines, these ones in parallel withtheir complex counterparts, and of Clifton-Pohl torus, to show that ourdefinition sheds a bit of new light on the behaviour of 'singularities' ofgeodesics in space-time. We also show that some geodesics, which 'end' atfinite time in the classical sense, can be naturally continued besides theirends. As a matter of fact, complex metrics naturally show a meromorphicbehaviour, or a degenerating one, so we shall study also this fact in detail.Comment: Completely revised version - mistakes fixed - 38 pages - LaTeX Please  note that the author is also known as "Claudi Meneghin

A Holomorphic Point of View about Geodesic Completeness