Fanaelialae
Legend
As I see it, if it's that rare why have it at all? The odds of a bad complication are only 1 in 6, while a good "complication" are also 1 in 6. These wouldn't affect the success or failure of the roll, but rather would simply add a little embellishment to the scene.I'd be cautious about even using a d6 for this - by that system you'd have a complication showing up on average every two rolls (2/3 chance of no comp. times 2/3 = 4/9).
I would suggest starting with 1 and 20 on a d20 having external effects on non-attack rolls and seeing if that's enough to cause the desired result.
Perhaps a d6 is a bit much (1 in 3 rolls will have something extra happen) but I don't think I'd go higher than a d10. I think the point of using such a system is that you want these little additions to happen with some frequency. If you use a d20, not only do you have to distinguish it somehow from the actual check (which may also have advantage), but it's also going to be fairly infrequent. Having the player roll an extra die for something that happens quite rarely seems like adding more complexity than what you'll gain from it. IMO, of course.