Perhaps one of the most famous time travel conspiracies of all time is that of the Montauk Project. In fact, the hit series, Stranger Things, even referenced it in the storyline. Several credible sources have come through to talk about their experiences at the now closed air force base.
Stories include being transported back to the 1940s, as well as being involved in very strange experiments and alien meetings. Disaster struck in the early 1980 when an experiment when horribly wrong. Many of the participants were apparently brainwashed in order to wipe their memory of what happened. Freaky.