Geodesic
Order Characterization of the Complex Field
Canadian mathematical bulletin, Tome 21 (1978) no. 3, pp. 313-318

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It is well known that the real number field can be characterized as an ordered field satisfied the “least upper bound” property.Using the idea of n -ordered set, introduced in [3], and generalizing the notion of l.u.b. in a suitable way, it is possible to give a similar categorical definition of the complex field.With these extended meanings, the main theorem of this paper (Theorem 7 in the text) is stated almost identically to the one for the real field. Any directly two-ordered field, in which the "supremum property" holds, is isomorphic to the complex field. 
Novoa, Lino Gutierrez. Order Characterization of the Complex Field. Canadian mathematical bulletin, Tome 21 (1978) no. 3, pp. 313-318. doi: 10.4153/CMB-1978-054-6
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     title = {Order {Characterization} of the {Complex} {Field}},
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