Swimming across the ocean?

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As of Alpha-1, this was possible, but the waters were not populated with mobs.

Generally though, you should probably expect that the only thing that will stop you, is the actual aquatic dangers.

Even if you can swim across the ocean, there'll still be a good case for single-player boats and aquatic mounts. Swimming is slow . In Alpha 1 I was miserable when I accidentally dismissed my aquatic mount mid-ocean (you couldn't summon them (or use any other action) while swimming in that build). If it's something you intend to do more than occasionally, you're going to want something faster, or which has some carrying capacity of its own. Not to mention that it's a lot easier to fish while sitting on the deck of a boat than while treading water.

There's a lot of disadvantages to swimming as a mode of transportation. People are going to want boats either way, even just for one person.