In multiple timelines where Subspace 2 is a thing, there’s an optional Chaos Emerald fetch quest to do throughout the game. It’s by far the most divisive part of the whole campaign. People who hate it think it’s too far out of the way and something where you need a guide to know what you’re doing, people who love it appreciate the additional dialogue, chase segments, and rewards. But if you want 100% completion, you have to do it.
The first Emerald is mandatory and part of the main story. Plot-wise, Geno and Ashley are looking for a “secret ingredient” (it’s mostly Ashley and Geno got dragged along) which is a red gemstone, so you do the level and at the end you get the cutscene where they find the red Chaos Emerald literal seconds before Shadow. Shadow gets pissed and chases after them, and it’s here you get your first chase sequence. Shadow will run after Ashley/Geno (whichever character you choose) and attempt to steal the emerald from you, and you have to activate Chaos Control at the right time to slow down time and give yourself a head start. Keep in mind that Shadow is the second fastest character in the game while Ashley and Geno both have mediocre ground speed. This first chase sequence isn’t that hard though since there’s very minimal platforming and Shadow won’t try to attack you, just catch you.
This happens fairly early on (~20 minutes into campaign) and after you get your first Emerald, the entire fetch quest opens up. To get the other Emeralds, you need to find three keys and use them to open up that respective Emerald’s vault, all of which are hidden differently. Sometimes you can find a key just by going off the beaten path of a level, sometimes you have to buy them, sometimes you have to do a minigame for them, and then sometimes the ways you get them are really stupid and don’t make any sense. The last vault in particular requires you to beat the Viridi boss fight by reflecting a reset bomb at her (this has to be the final blow). And if you don’t have all of the keys at this point or god forbid that Shadow catches you after you open the vault, OOPS! You have to do that boss again. This one in particular is what a lot of people hate about the fetch quest since it’s not at all obvious that that’s what you need to do to get the last vault, and the platforming in the chase sequence is kinda bullshit (if you chose someone with a tether recovery like Joker or Hat Kid, good luck).
It is technically possible to do the entire fetch quest right away in one sitting, but it’s not easy. None of the Emeralds are locked behind events in the main story so you can get all of them right away, but doing so requires you to go into areas that you’re wildly underleveled for (unless you’re in New Game +) and you’ll most likely game over before you can even get to where you need to be. The chase sequences get harder and harder with every Emerald, as there will be obstacles and enemies in your way and Shadow will try to attack you. With Chaos Control at your side, these chase sequences are certainly doable, but each character’s Chaos Control timer is different (for example, Bayonetta and Shulk have the longest timer, Snake and Bandana Dee have the shortest) And if you fail at the chase sequence, you get booted back to whatever the last checkpoint was, which sometimes can be right in front of the vault and other times can be a ways back. Combine this with the difficult levels and weird requirements for getting the vaults, this can get very tilting depending on who you play as.
Obviously, you’re not supposed to get all the Emeralds in one go, and it’s a lot easier if you do it as the main story progresses as opposed to doing it right away. People do it anyways because there’s an achievement for getting all the Emeralds and a bunch of other achievements associated with it. Also people are insane.
Aside from that, there are two main rewards for doing the fetch quest:
1) Once you get all of the Emeralds, Shadow will join you and become playable. Unlocking all of the characters is required to get the good ending, so this is kind of a big deal.
2) For the final battle against True Tabuu, you have the option to use the Emeralds to go super for your first character/stock. You get a major power/speed/durability increase and a better recovery. Only a few characters are able to go super, but among the ones that can, Bayonetta is easily the best choice. General power increase aside, the Emeralds buff the HELL out of Witch Time, making it last just as long if not longer than it does in her home series. This means that if you’re good with counters, you can beat True Tabuu, the final boss, in less than a minute. Bayonetta’s Witch Time is also a lot less picky about timing than Jeanne’s or Shadow’s Chaos Control, so she’s the better pick. This is how most people defeat True Tabuu on Devestating since he’s a damn hard final boss otherwise (in a good way!)
Other than all of that though, a lot of people (mostly people who don’t do it all at once) really like the fetch quest because they enjoy going off the beaten path to find they keys/vaults and the extra minigames associated with them. Also, depending on who you open the vault with, you’ll have some extra dialogue with Shadow based on the character you choose. It’ll also become fairly popular to do self made challenges of the chase sequence, most of them involving not using Chaos Control. For example, people will use tether grabs to keep Shadow away, Arle can lock Shadow behind walls of Puyo combos, and Sonic can just straight up outrun him as long as you don’t mess up your platforming.
TL;DR: Long, optional sidequest that a lot of people love and a lot of people hate. Required to unlock Shadow.