The notion of infinite prime in a ring with identity, defined in the first author's memoir "Finite and infinite primes for rings and fields" (A.M.S Memoir #68), is studied in fields. Extending results of R. Baer and D. W. Dubois, each infinite prime P of a field F is shown to determine a complex place φP of F such that φP{P) is the set of nonnegative reals in ΦP(F), and is an infinite prime of φp(F). The collection of all infinite primes P of F determining the same φ and φ(P)9 is shown to be describable, in an almost purely multiplicative way, in terms of certain groups determined by φ and φ(P). Using these theorems a notion of completion of a field at a finite or infinite prime is given, generalizing the classical notion for the prime divisors of a number field. These completions are characterized as certain linearly compact fields and are shown to be in general unique only when P is real (i.e., φP(F) is contained in the reals). For a fixed prime P of F, the set of elements of F which are squares in every completion of F at P is calculated. A characterizati on of number rings is given and examples of pathology in infinite primes are indicated.