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If an independent character has joined a unit that comes under the effect of a psychic ability--let's use Doom as an example--he is also affected by the power. If he then leaves the unit is he still affected?

Are there any powers like this that last for more than one player turn?

Eldar Fortune and Guide could work like this. The Farseer in question could cast either of them on a squad he has joined at the start of the turn. He then leaves the unit in the movement phase--does he retain the effects of the power?

I'd say no in this case, since he is no longer part of 'this unit' that he cast the power on. In other cases, it'd probably depend on the wording.

... intresting how about a unit thats been doomed the IC leaves the unit and charges in to CC combat is he still doomed?

...I would have to say that its the unit that gets the effect ... and any IC joinging the unit also gaines the effect ... any leave lose it

Well, even if the unit charges into CC the IC is not part of the unit as long as the cc goes. Just when it comes down to the outcome of the cc the IC is part of the unit again.
Seems to be a little loophole in the rules or at least something to discuss about.

In Fantasy, if a unit with a character is subject to *any* remains in play spell, and the character leaves the unit, the casting player may choose whether to move the spell onto the character (relieving the Unit) or leave it on the unit (relieving the character).

Seems sensible enough.