The 11th Doctor frequently wore a fob chain with his waistcoats, and, as best I could tell, always wore it with his “anniversary” waistcoat.
He also wore the fob chain in the corresponding publicity photos for the anniversary special.
Matt Smith usually (perhaps always) wore the fob chain with his “anniversary” waistcoat during the behind-the-scenes footage on the Blu-Ray special features.
Of course, since Eleven was wearing the fob chain with his “anniversary” waistcoat when he regenerated, Pater Capaldi briefly wore it at the beginning of his first episode, as well.
Curiously, in this publicity photo, Peter Capaldi wasn’t wearing the fob chain … however, considering both the shirt and waistcoat were partially unbuttoned, I think this was probably more of a “casual” photo taken either between or after filming (as opposed to an in-universe reflection of the character’s wardrobe).
Here are some detail shots of the fob chain.
In this particular photo, one can see a bit of the center clasp.
