In the beginning of the series (both 2003 and Brotherhood) Cornello takes way Ed's pocket watch in order to prevent his circle-free transmutation (which is later revealed to be pointless). From this, it was implied that the pocket watch has some sort of effect on a State Alchemist's transmutations. I seem to also recall that in the 2003 series, when Ed became a State Alchemist and was given his watch, he was told it could boost his alchemy.
However I don't recall any instance in either series where this was the case. It's not used as a mobile circle as each combat State Alchemist had their Transmutation Circle with them (Roy's ignition gloves, Gran's gauntlets, Armstrong's metal knuckles, Kimberly's own hands) and Shou Tucker was never shown to use his (and I doubt that he used it for producing the Alexanina Chimera because he was capable of doing this from the start, having done a similar transmutation before becoming a State Alchemist).
I recall at one stage the chain was used to bind someone, but what I would like to know is if there was any situation, in either series, in which the boosting capabilities of the State Alchemist pocket watch is shown to have benefited an Alchemist's transmutation (without the addition of Red Stones/Fake Philosopher Stones as per the Lior Rebellion in the 2003 series).