What do you mean is broken!?

Fine. Every time you change classes, you forfeit all exp and start from scratch.
- This is an experienced DM.
  
  Truly experienced is to only drop one level with each change. Then you get to watch them try to weigh cost/benefit.
- Nah, random class two of which are town idiot and handless deaf mute bard.
  
  handless deaf mute bard.
  So, fart musician?
  
  handless deaf mute bard
  Then how does viscous mockery work
- I was thinking every time you change class you lose 1 level, to a minimum level of 1. After a long rest you regain your original level, and 1 'charge' of your next class change (you're level 8, you changed classes three times before rest, you rest at level 5 wake up at level 8 (whatever class you want to be, sure), but when you change class today your new class is only level 6. Maybe after level 8 you recover 2 charges per rest.
  Edit, fixed my after 10 bonus, changed 10 to 8 to be more in line with when other classes get their bonuses
  
  Seems like a new to a class person wouldn't know any skills. Any maybe wouldn't have refreshed spell slots on every change. And would still need to change outfits manually.
  
  Fair idea, but then I'd be condoning this decision.
- Would it be close enough to balanced if all the classes simply have their own level and xp? So you effectively can get the abilities of every class at level 1, which I don't think would be overpowered. You'd also be effectively penalized for staying too long in any class except your main one, cause you'd be earning XP for an alt class.
  
  Each class's first level features are at least as potent and useful as a Feat. Getting 13 of them off the bat, with a combined like 15 cantrips and first level spell slots, rage, two different unarmored defenses, a hundred first level spell slots, Eldritch Blast... Early game, you'd be wildly overpowered, but late game you'd be struggling to keep up with your more specialized team mates

That's like changing your major in college from music theory to advanced physics in one turn
- I did that exact major change, lol. Took more than a turn though.

If you can only do this on long rests it basically doesn't affect balance, right? You don't ever get extra resources, just different ones.
We let somebody do this in our 4e campaign and it was fun. They had like 4 different character sheets that all represented the same character, they just picked one for each adventure. It let us vary our party composition without having to introduce new characters to the narrative.
(The conceit was that they were experimented on by a mad mage who imbued them with elemental essences in a failed attempt to create a pan-elemental being; instead they manifested one element at a time, and each of their forms had its own abilities. Mechanically their race was genasi, with a different elemental aspect for each class.)

Monkey's paw it. Give them a poison first session that makes them permanently lose control of their shifting.
Make their class random every dawn. For for chaotic madness, make their class randomly shift every 6 seconds at the beginning of their turn. Or make it change to the least useful class for the particular situation. Or maybe the player doesn't even know which class their charcter is at any one time and must decide what to try before they know if they're even the right class to try that. ("You gesticulate and make a constipated face and feel at one with the forces of... rolls dice... smashy smashy. You've forgotten what a 'magic missile' is.")
- A random class every day actually sounds like a fascinating challenge. It'd be such a pain to gear up, since you never know what you'll be proficient with on a given day lol.
  
  Hear me out. A whole party full of characters like that. They'd just need to collect a set of gear for each class and then trade things around every morning.
  Honestly, I kinda want to play in that campaign.

Maybe im a not a fun dm for everybody but I be usually require classes from a published source. I barely have time to play DND as it is and I don't want to worry about balance issues or rules that haven't been play tested. Even of playing DND for 8 years I wouldn't feel comfortable doing it. I just wanna play DND and not have to worry about hurting anyone's feelings on their pet project.
- Nah this is the way. Most peoples homebrew class or race or weapon or whatever is shit.
  
  Pretty much. If the DM has a desire and you can work with them on the class that's cool but it shouldn't be expected. The DM puts the time and effort for the session and players should understand that.

The class he can shift between? Murder Hobo and Paladin.

So basically the warrior of light from FFXIV
This checks out

I see most of the chat isn't familiar with the 'chameleon' prestige class from 3.5. It was super underpowered but it seems like you all would think that means updating it would be a good approach to handling this player.
My wife played a changeling chameleon one campaign, and while the rest of the party blew her away in combat, in RP she was pure gold.

The same people who you used to play with as a kid where they somehow have the power to have all the powers

I think this would be pretty cool with some more rules behind it. Like they can shift into another body within range and literally gain everything that creature has, but it has limited range and a save to resist being taken over based on the hitdie, level, both, etc.
- we can go further: what if the ONLY way they have to progress is copying another creature? and they can't just go back, they'd have to find the creature they previously copied to do so again.
  now they can become powerful sure, but most powerful creatures aren't something you can just walz into a town as, and every time you change form you have to abandon your previous powers.
  
  Be pretty funny to make every encounter normal for the other players, but basically a puzzle game problem for that one dude. 🤣

I thought it is a political meme.