If you remember Scoobie Doo, every plot turned around the notion that there was some mystery that seemed to have something supernatural at its root. However, in the last scene, the spectral figure would be unmasked as old man caruthers who'd invariably say "I'd have gotten away with it had it not been for those meddling kids."
It was very important that these stories have no actual supernatural explanation.
Then there was Magnum PI wherein you'd have plots
wherein a beautiful woman would be the spitting image of the long dead victim of some decades-old crime. There would be broad hints and possibilities that perhaps there was something supernatural going on. Maybe the ghost of the dead girl gave Magnum the vital clue, or maybe it was a coincidence. There was both a naturalistic explanation, and there was a supernatural one, too.
It was very important that these stories have a naturalistic explanation for everything except for one spooky ambiguity introduced in the last scene. Though you might see t
he ghost in the last frame, there'd be no proof.
Castle is a TV show that does this a lot. You can see several episodes wherein something spooky, science fiction, or horror related might explain the crime. Rick Castle is all about wild about jumping to the supernatural conclusion, and Kate Beckett spends the show shooting him down and coming up with a reasonable explanation.
Castle is not the Sarah Connor Chronicles. Nor is it the Dresden Files. This places constraints upon the writer if s/he wants to avoid jumping the shark.
One of those constraints is to maintain ambiguity. You can't prove definitively within the story that the murderer or the red herring actually was from the future.
This means you can have the putative time traveller disappear from Castle's view, but you cannot allow Beckett to spill coffee on a letter in exactly the same pattern as shown in the picture of the letter from the future. If the pattern varies, you can plausibly maintain the photo was a coincidence. And if it varies, you can use a butterfly effect argument to maintain the time travel was real.
Clearly, I should be writing more and watching TV less.