The Hebrew and Greek that is translated as “swear (an oath)” or “vow” is translated as “God sees me, I tell the truth to you” (Tzeltal), “loading yourself down” (Huichol), “to speak-stay” (implying permanence of the utterance) (Sayula Popoluca), “to say what he could not take away” (San Blas Kuna), “because of the tight (i.e. “binding”) word which he had said to her face” (Guerrero Amuzgo), “strong promise” (North Alaskan Inupiatun) (source for all above: Bratcher / Nida), or “eat an oath” (Nyamwezi (source: Pioneer Bible Translators, project-specific translation notes in Paratext).
In Bauzi “swear” can be translated in various ways. In Hebrews 6:13, for instance, it is translated with “bones break apart and decisively speak.” (“No bones are literally broken but by saying ‘break bones’ it is like people swear by someone else in this case it is in relation to a rotting corpse’ bones falling apart. If you ‘break bones’ so to speak when you make an utterance, it is a true utterance.”) In other passages, such as in Matthew 26:72, it’s translated with an expression that implies taking ashes (“if a person wants everyone to know that he is telling the truth about a matter, he reaches down into the fireplace, scoops up some ashes and throws them while saying ‘I was not the one who did that.'”). So in Matthew 26:72 the Bauzi text is: “. . . Peter took ashes and defended himself saying, ‘I don’t know that Nazareth person.'” (Source: David Briley)
See also swear (promise) and Let your word be ‘Yes, Yes’, or ‘No, No’.