Does Serana keep following you after Dawnguard is over?

She will become a normal follower. You can part ways and enlist her again.

Serana is a little buggy as a follower. This can be used to your advantage, however, if you would like more than one NPC follower.

There are points during the Dawnguard quest line where she stops following you, and others where she starts following again. If you pick up another follower while she isn't following, you can keep them when she starts following again. You can have Serana continue to follow you once the quest line is over, and you will have 2 followers. You can even dismiss the second follower and pick up another instead while Serana continues to follow. Just never dismiss Serana and you can do this.

Probably the best point to do this is after you have defeated Arch Curate Vyrthur and before talking to Knight Paladin Gelebor so you can be given Auriel's Bow. During this brief window, Serana will stop following you. You can fast travel away and pick up someone else(Lydia and J'Zargo are just two examples) and take them with you back to the Forgotten Vale. After you talk to Gelebor, Serana will start following again but you'll retain the services of whoever you traveled away to get.

If you dismiss Serana after the Dawnguard quest line is over, you may run into a bug where you can never have her as a follower again in the same play through.

I have personally had Serana and another follower(mainly my wife Lydia) following me simultaneously for a while after the DG quest line on a few occasions, but Serana's penchant for necromancy and careless use of chain lightning have caused fights between the two(easily fixed with a dual cast of Pacify from a dragonborn with 100 illusion skill and most of the perks).

Serana can make for an interesting follower, but I find that the pros of such an arrangement are generally outweighed by the cons. Usually I don't need a follower because of highly perked weapon skills with well tempered(and enchanted) weapons and armor in addition to heaps of stamina, a combo of Extra Pockets and the Steed Stone, and a number of powerful custom fortify carry weight enchants I can switch to when I start running kind of heavy( mostly happens when going through a dwemer ruin or nordic ruin).