Most common water filters are actually pretty useless. People buy them thinking they'll filter out stuff like lead and fluoride but they don't. At best they'll filter out visible particles.
Distilled water and reverse-osmosis filtered water is the closest to pure you can get.
Why would you want it that pure anyways? Many minerals in water and the flouride in small amounts is actually good for you.
There's no need to drink fluoride as swishing with tap water achieves the same intended effect, and the amount of minerals in tap water is far too inconsequential to confer any possible health benefits, but the contamination that could not be removed (trace heavy metals, unmetabolized prescription drugs) could cause deleterious effects through chronic ingestion.
Cept that there are studies that show that distilled water will leech the plastic from the container if left too long, and there might be issues with it leeching minerals from your body since it is missing so many.
Most "studies" on distilled water are pure hogwash perpetrated by the quack Dr. Mercola. The only thing that's true about distilled water is that it obviously cannot replenish lost electrolytes (but tap water is just as ineffective). ---