I think the only thing close to what you want, until SE decides to implemeting an in-game solution, is by going to the bottom of the Chat window and tabbing over to the Battle tab. However, that replaces your entire chat window to battle info. Out of PvP, it has shown when buff muipliers proc on my positionals, as well as numerous details of my damage output, when I was hitting a striking dummy as a MNK for example. I've never used it out of that random time I accidentally switched to it and it showed that info.

Thats the other problem that certain things like that are already in game but they are hidden and not well-known.