No. It’s not something one “has”. It’s a complex situation or drama a boy goes through at a certain stage in early childhood. Psychoanalysts (Freudian psychologists, whose numbers are relatively few nowadays, as Freud’s theories have been partially discredited) use this supposed developmental stage, which I think is farcical to begin with, to explain both normal childhood development and abnormal adult behavior; a fixation (part of the libido getting stuck) at this stage could result in sexual deviancies (both overindulging and avoidance) and weak or confused sexual identity, among other things.
Laymen might use it more broadly, without TOO much of a stretch, to describe men who are attracted to women who resemble their mother, I suppose…
Any usage beyond that would just be ridiculously wrong, similar to the now ubiquitous misuse of “schizophrenic” to describe what should be called “multiple personality disorder”, and so on.
The Oedipus complex involves sexual awakening by the boy (age 2 or 3-ish), sexual attachment to the mother, and rivalry with the father over this lover. Upon realizing the sexual nature of the boy’s interest, the mother will forbid him to touch himself, and threatens him with castration if he doesn’t stop. She may say the father will carry out this castration. The boy, having seen girl’s genitalia, realizes that his penis can indeed be cut off, as girls are proof of this. The threat is too great, and the boy represses his sexuality, so that throughout the rest of childhood, the child is an asexual being. The boy yields to the father, becoming an obedient son, rather than a rival. In fact, he identifies with him, in order to deal with the threat, and this is how his masculine identity develops; he also becomes repulsed for ever after by the thought of his mother’s sexuality. The whole event thus drives the development of the superego, incest taboo, masculine identity, repression of sexuality in children, and so on.
The term Oedipus or Oedipal Complex is used to describe this developmental experience; it is not a label for adults or adult neuroticisms. We would not say “he has an Oedipal Complex”. Rather, everyone has gone through this during early childhood. I think the term “complex” is misunderstood by laymen here, as the word can be used to describe a personality problem in phrases like “inferiority complex”.
If the events go badly, you supposedly might end up with fantasies about your mother, hatred of your father, an overly masculine or feminine personality, homosexual or bisexual, and so on. I’m not an expert in psychoanalytic theory, but this should suffice to give you a rough idea of what Freud thought. I think the whole theory here is a bunch of hooey, personally.
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