NP-complete promise problems?
There's two trivial answers:


No. For a problem to be NP-complete it must be in NP. To be in NP it must be a decision problem and promise problems aren't decision problems (they don't have to answer Yes or No on inputs outside the promise).
Yes. If you want to say informally that something is NP-complete (usually meaning "there's an obvious equivalent decision problem that is NP-complete"), then you can reformulate any decision problem as a promise problem just by taking the promise to be the set of sensible inputs. For example we can make a promise version of Dominating Set by taking the promise $L_{YES} \cup L_{NO}$ to be the set of all simple, undirected, unweighted graphs (so if you give it an input that's not a graph, it doesn't have to do anything in particular).


So there's no list of NP-complete promise problems because either none of them are (the strict answer), or you can just take any NP-complete problem and make a promise version.