This is an effective measure used on other sites. Usually it results in account merging, but as this isn't as effective on Reddit-like sites it'd probably be better to just flag them as "aliases" that are unable to vote on posts, and invalidate the duplicate votes.
This would solve one particular problem we have here.
[–]wizzwizz4[S] 1 insightful - 1 funny1 insightful - 0 funny2 insightful - 0 funny2 insightful - 1 funny - (5 children)
[–]Tom_Bombadil 1 insightful - 1 funny1 insightful - 0 funny2 insightful - 0 funny2 insightful - 1 funny - (4 children)
[–]wizzwizz4[S] 1 insightful - 1 funny1 insightful - 0 funny2 insightful - 0 funny2 insightful - 1 funny - (3 children)
[–]Tom_Bombadil 1 insightful - 1 funny1 insightful - 0 funny2 insightful - 0 funny2 insightful - 1 funny - (1 child)
[–]wizzwizz4[S] 2 insightful - 1 funny2 insightful - 0 funny3 insightful - 0 funny3 insightful - 1 funny - (0 children)