There's a really widespread myth about Peter I's trip to Europe which states that he was replaced by a stranger, possibly Alexander Menshikov's puppet. The scientists who first did research on this topic weren't historians but mathematicians, the author of New chronology Fomenko and Nosovskiy, therefore it is not to be trusted.
However, is there any actual historical evidence that would support those claims or is it completely groundless? I don't have experience at working with historical sources so I would really appreciate any help regarding this question with the sources. Thanks in advance.
The very short answer is "no". A lot of these aspects of New Chronology, the elaborate discourses about conspiracies and how things were 'faked' are an odd sort of example of post hoc ergo propter hoc playing out within the thinking of an individual to maintain logical consistency with the core argument. If you accept the core argument of the 'fake time' and the falsification of history then other claims necessarily follow to justify abnormalities, especially abnormalities that would screw with his "parallels" of historical dynasties.
It is worth noting that Peter comes along a century after an ACTUAL fake Tsar, Dmitri, who claimed to be the, probably assassinated, son of Ivan Grozny. Thus these things are not without precedent, making the absence of recorded evidence all the more startling.