I have seen this done before - it was the result of an in-game curse, so slightly different than an innate trait of the character, but functionally equivalent.  Same base stats, but with different classes and such, and a dice roll to determine what personality would be used.

It was a fun idea in theory... but quite unfun in actuality.  Never quite knowing what character you were going to deal with (which could make both roleplaying and planning hard) decreased the enjoyment of everyone at the table.  Additionally, the player had to internalize and keep straight two character sheets--that could easily result in them playing more slowly, particularly if they got confused over what abilities went where or what abilities the characters even had.

Overall, can it function?  Yes.  But from personal experience, I would highly caution that this is one of those ideas that sounds much better than it actually is.

In addition to what Caerwyn has said...

as someone who has Dissociative Identity Disorder, I also want to point out that it is very hard to play such a character appropriately, and without resorting to stereotypes, unless you yourself live with the disorder. 

In short, unless you do a LOT of research, you will not be able to run it appropriately. And you will cause far more offense for those who actually have the disorder. This kind of thing comes up a lot, and the answer is "No. don't. ever."

Having a separate voice in your character's head, such as a Warlock's patron who is bound to them and trying to influence their decisions? Sure. go nuts.

But DID? no. stop it. It's far more effort than its worth making *characters* with it rather than caricatures or stereotypes.

As Neutralista said, making a character with this sort of real-world disorder is asking for problems. And as Caerwyn_Glyndwyr said, the mechanics of it are prone to become difficult, janky, and hard to manage.

Now, that all said, here are a few backstory options to give you an alternative to DID on the character whilst achieving the effect you're after:

• When you were a child, your sibling was killed, and your innate magic saved their soul from the evil necromancer. You always shared everything - now you share a body.
• You were an Ettin who wished to be transformed into a Dwarf. You never stopped to consider that they only have one head.
• You picked up a sentient magical item which originally intended to take over your body. Luckily, you had an amulet of protection, so it didn't succeed. Completely.
• You are a barbarian who channels the spirit of their father when they rage. Their warcry is some piece of ancient anger about something called a "Thermostat".

Mechanically, you could do the following:

• As a barbarian, when you rage, you adopt a different personality.
• Ask the DM if you can contract Lycanthropy, giving you a second persona. You may or may not have control of it!
• Multiclass into contrasting classes (EG Melee fighter and ranged Sorcerer). Whenever you make an action, you have a different personality taking the helm to control the character. This way, it's purely roleplayed.

One additional issue with this concept (which people haven't mentioned because being insensitive and not having fun are probably far more of an issue) is that balancing it is an issue. You will either have 2 character you swap between, but the HP is the same, or you have no control over what abilities you can use, which means you're underpowered as a multiclass. As well as this, if your character has two classes, then you may get other players feeling like you're stealing the show somewhat.

So in my opinion (summarised):

• Roleplay it, don't make mechanics
• make a fantasy reason for it, not a real-life one

C_G, Neutralista and Thoruk's responses should be pinned for whenever this subject comes up

People with DID exist in the real world and they don't have any special powers.

If what you're after is an aesthetic shift to represent being "taken over" by your powers, a la Percy in Vox Machina, Raven in Teen Titans, or something like that, then look for options that use a mode shift. Barbarian, Glamour Bard, Bladesinger, Undead Warlock all come to mind. Rune Knight Fighter also, kind of.

Quote from ChoirOfFire >>

People with DID exist in the real world and they don't have any special powers.

If what you're after is an aesthetic shift to represent being "taken over" by your powers, a la Percy in Vox Machina, Raven in Teen Titans, or something like that, then look for options that use a mode shift. Barbarian, Glamour Bard, Bladesinger, Undead Warlock all come to mind. Rune Knight Fighter also, kind of.

 Lunar Sorcerer also fits very well. 

There are several species in Monsters of the Multiverse which are also worth looking at: the Changeling especially, with its multiple personas, but also the Eladrin’s seasons and the Shifter’s shifting. 

Quote from worldscribe >>

They'd have different classes, different proficiencies, different languages, and at least not start off with knowledge of their other personalities. 

 This just doesn't work in practice. One of your personalities has used a Rage - how does that translate to resources your wizard personality has spent? Or do you just expect to have access to the full resources of both classes every day?

Stick with one character and roleplay it like you'd roleplay anything else.

Astral Self Monk, Stars Druid, and Spores Druid also fit the concept I described.

Phantom Rogue gets the ability to swap proficiencies sometimes, if that's more important to you.

I agree with those stating not to craft the character as actually having DID.  

There are a few ways to make something that might be interesting without co-opting a disorder real people deal with.

I would instead flavor it as two souls inhabiting the same body, perhaps due to some kind of curse or a resurrection spell gone wrong, perhaps a sentient Clone that awakened and become their own person before their creator died and their soul transferred to the Clone body causing both of them to share it.  Something like that.

Running it, I agree with what the OP said.  Have the same stats for both characters, but different classes and languages and the like.  I agree that it could get confusing for someone to have two entirely different character sheets to contend with, so suggestions like others have made like having the alternate personality emerge when raging or activating a certain subclass ability would be good.  Another way you could do it (DM allowing of course) would be having the same class but two very different subclasses.  That way the base mechanics are the same but the flavor is very different, say a Thief rogue vs a Phantom rogue.  Or an Echo Knight fighter vs a Samurai fighter.  

I disagree with the idea of running two classes or two subclasses. There's enough tracking to do as it is. And it frankly isn't equivalent in power to have two different sets of features, even if you can only use each of them half the time. Or less! Like even if you only got a third of the spell slots and a third of the uses of Rage, for example. Or like, a quarter of the Ki and a quarter of the, idk, sorcery points. It's just hella complicated to try to evaluate the power level. And if you get it wrong, somebody's going to feel useless.

Edit: By level 20 my Paladin had an entire sheet just to track resources. Hit dice. Lay on Hands. Paladin spell slots. Racial spell uses. Ring of Spell Storing spells. Channel Divinity uses. Divine Sense uses. Bag of Tricks uses. Wave, the legendary magic trident, had like 2 or 3 different resources to track if I recall correctly. I had a charm of animal friendship or something with a couple uses. Necklace of Prayer Beads spell uses. Spell components that get consumed! Gold!

That's true that it could become complex.  That could be down to the player, their familiarity with different classes/subclasses, and which classes, subclasses they would want to play.  Just splitting a specific class and doing two subclasses, something like Assassin Rogue and Thief Rogue would be mind-numbingly easy.  A more basic fighter as one of the classes would mean when you switched to them there's really not that much more going on you would need to keep track of.  Another thing would be frequency of switching personas.  If it happens maybe once per game session then it isn't really that much more to keep track of, especially because most of the "keeping track of things" for many classes has to do with resources used in combat.  If we are talking about purely social situations then switching back and forth would be entirely on the player just being able to keep each personas desires, fears, etc in mind (besides some abilities that can be useful in social situations).  

All in all, I think it is entirely doable, but it would be on the player to keep track of their stuff and to choose two compatible classes/subclasses (if that is the direction they were going with it).  

As the OP said it would be something like "Moon Knight", that's really just one commoner whos persona would be something like a Paladin or Warlock.  If the alternate persona is even just a Champion Fighter, that's pretty easy to keep track of.  Depending on what you choose for Fighting Styles, D&D Beyond does most of the heavy lifting for you.

Moon Knight gets to fill two roles in his story (at least in season 1 of the show!). He's the smart guy and the muscle, just at different times. That works fine because there's, what, two other people in his party at most? Usually fewer. He's not stepping on anybody's toes.

Just something to be thinking about as you move forward with the idea.

Quote from ChoirOfFire >>

Moon Knight gets to fill two roles in his story (at least in season 1 of the show!). He's the smart guy and the muscle, just at different times.

That actually wouldn't be hard to do mechanically without making two separate characters: barbarian with a background like Archaeologist (if we're specifically talking Moon Knight) or another one with "bookish" skills, maybe adding in something like the Skilled feat, and then role-playing Rage as switching which persona is in charge. You could even RP a blown skill check after a Rage as the muscle not going away at inconvenient times