The Gyrojet (Handheld Rockets)
I'm surprised this wasn't mentioned yet, but have the gun could fire bullets with rockets behind thim. This isn't even new, it has been around since for at over 50 years. It is called the Gyrojet, and wasn't simply a prototype, but a full production line of firearms including rifles, pistols, underwater firearms, machine guns, and derringers. There were over 1000 of the pistol alone made.
While you don't expand on why you want them, I'll list several advantages.
The speed of the gyrojet was just around the speed of sound. Those that went below were reported to have made only a 'hissing' noise. With improvements to the technology and modern/scifi level manufacturing, it could be possible to make it even quieter or have no noise with improvements to the rocket nozzle. Or, you could possibly have the sound directed directly back to the shooter so no one else can hear it (Some stealth military jets direct noise/heat/light in as focused of a beam backwards as they can to avoid detection.)
With more development, the firearm might not even need a chamber (as traditionally used), so it could be possible to have it shoot more like a toy gun where it's discharged directly from the top of the magazine into the barrel. If this happens the striker, firing pin, or other ignition system could be operated completely by the trigger. This would make the firearm even quieter, as one of the loudest parts of a suppressed firearm is the action cycling. This could be improved on even more by having electronic ignition of the rocket. If this happens there are no moving parts beyond the rounds themselves.
Assuming you had electronic ignition, there are no moving parts to jam. The rounds are 'ejected' when they exit the barrel, so the only moving part is the magazine. Obviously this bodes well for reliability. Of course electronic could be augmented or made more reliable in many ways. They could even have the ingredients for a battery inside them and when they move up in the firearm they are 'primed' somehow by rotating or pressing on the back half of the round to mix them and you have a brand new battery. Or just have a capacitor that doesn't discharge over time integrated into the rocket. Caps get pretty small, could get smaller in the future. Maybe even have regenerative firing where the heat or gas pressures of one round charges up the weapon.
Or even have 'wireless' ignition. Have them stored in an RF blocking case/magazine, and they only reach a window where RF can reach them when it's 'primed' or ready to fire. Heck, you could even block RF from all but one direction. Send a pulse of RF that's enough to trigger a spark gap in the round and fire it. This allows it to be fired completely remotely with no gun at all (you only need the round).
Even if they are mechanical, only having a trigger activated firing pin is substantially simpler than all modern firearms, even revolvers, bolt actions, and single shots. Not having to deal with chambering and ejecting a round is fantastic.
Maybe even have both. Electronic ignition, with a primer as a backup. Have the trigger set off the electronic before the mechanical striker releases. No settings to touch, same action of pulling the trigger fires whether there's battery or not.
Cycling issues are also a very common cause of failure in semiautomatic firearms. Many of these will not exist if you don't have any moving parts on the firearm. I'm not sure it would be helpful to explain how the various failures happen depending on the action of the firearm, since we can assume this kind won't even have an action.
Since there is no explosion pushing the round against the firearm, there is (practically) no recoil. The rocket is self-propelled. This can allow faster follow up shots with better accuracy. Additionally, this is more practical for low/no gravity situations. A regular bullet will push backwards with equal and opposite force. A rocket will only push backwards minimally while it is close by, and is self-propelled past that point.
Regular bullets have an arc, because they are effectively 'thrown' from the gun at high speeds. The farther they shoot the higher the arc has to be. This also makes aiming long distances much more difficult, as you need to calculate bullet drop.
This could even have applications in some odd scenarios. Underground tunnel system where there are man-sized tunnels miles long? Might not be possible to arc a bullet without hitting the ceiling, but that wouldn't be a problem for a rocket.
Being in the future it's not at all unreasonable to think there could be some logic in the bullets. This already exists, albiet in the prototype stage. DARPA's EXACTO round is one of two competing prototypes. It is capable of being shot out of any existing rifle chambered in .50 BMG. Considering that this already exists it's not unreasonable to think that the technology could be applied to handheld rockets too. It could redirect to 'miss' a friendly, or help hit a specific part that's being aimed for. Nonlethal could aim for an adversaries weapon to disable it. Being the future, most other non-ai weapons personal weapons like lasers likely won't be able to add much intelligence. Maybe even have it shoot around corners. The possibilities are endless.
One thing I was thinking would be interesting would be to use them like a tiny little mirv. They can use the rockets at the end to overcome wind resistance and go in at full speed. Just have a sattelite drop a few thousand from space and they can do whatever they need to on a large scale. Take out all power, phone, or fiber lines, even if they are buried? Shouldn't be a problem.
The underlying technology could possibly even be used for non-weaponized purposes. In my head I'm picturing a nailgun with intelligent rounds. Fire from across the job site and it nails itself up. Have some self-propelled bullets (They are just rockets, remember?) hold up a board while others nail it in. They only need to hold it for a couple of seconds, it doesn't seem that unreasonable. Okay enough with the tangent...
Regular Bullets start off at full speed (after leaving the barrel) and decelerate the entire time. This also limits the effective range of subsonic ammunition.
Rockets can continue to accelerate throughout the flight, or even maintain speed. This allows a distance of travel only limited by the amount of fuel it can carry. Combine this with intelligent rounds and you could fire them from anywhere in fuel range, even hundreds of miles away (assuming you had a high tech dense fuel). They can also continue to accelerate after hitting the target, allowing them to hit multiple targets or possibly go through several layers of protection, as they can re-accelerate every time something slows them down.
This also allows more energy transfer into the target. The rounds can be as heavy or dense as needed, and the rocket can maintain just below subsonic speed. This, again, allows a stopping power only limited by the weight of the bullet. This may not be an issue at all for vehicle mounted weapons. An extremely dense very small round weighing a couple of pounds would be trivial to carry hundreds of on even a motorcycle, and a larger or military vehicle could carry thousands. This would be an extraordinarily deadly weapon. Even against armor you can combine this with 'Intelligent Rounds' above and it could find the weak points of armor on a tank or building and attack those. Since it's not really feasable to put extreme armor on every single portion of something, you won't need nearly as strong of an armor piercing round. Even tanks and "IED resistant" platforms use different armor on the front/sides vs the bottom. An explosion is different than an armor piercing round (A chunk of dense metal being thrown very very fast).
Simpler, Lighter, and Smaller Weapon
Since there are no extreme pressures on the firearm, you can make the weapon out of nearly anything that could withstand the brief temperature of a rocket passing through the barrel. If we assumed electronic ignition, you could have a magazine and a barrel made out of high-temperature tape or plastic. Not that I'd want to rely on it being accurate, but the possibility is a semi-automatic version could possibly be made in seconds out of household materials, with no tools except tape needed.
Not needing any high-strength materials also lowers the weight of the weapon. This is always good.
Smaller also unlocks some interesting possibilities. You could have a single shot version with no weapon at
all, only the bullet. If you can fit the bullet somewhere, you can fire it accurately from there. You only need a magazine to house other bullets, reload, things like that. You don't necessarily even need a barrel. Imagine having only a magazine, and a button on front that fired it. You could easily hold dozens more rounds on you than you can now. This also helps even more with the reliability aspect. If the 'magazine' is a self contained gun that fired rockets, you could have 3, 4, or 5 backups on you at any time. If there is a problem just drop it and grab another, like you were reloading. This is sometimes humorously called a New-York reload. You could have door knobs that can fire out electronically, or set traps, or have thousands of guns that fire simultaneously (even in 3d! Just have rounds behind each other that fire at the same time). Since each bullet is basically a gun by itself they can be fired without any supporting infrastructure (or even wirelessly). This effectively even allows you to use them to basically shape themselves into a larger object. Imagine them as a bunch of tiny little 3d pixels. I believe the mythbusters did something similar to paint a picture with paintball guns.
Last thing I'll say
Most bits of technology have been available or working for years, and much of it was developed decades ago. It's not a stretch at all to think that some extra development or near-future tech could make it very useful for a long time or it could even become the standard firearm of the future.