Every good character should have an arc.
Okay, no, that's totally not true, but just like any of the "rules" of writing it's a pretty effective staple of storytelling. If you want to make a character compelling and their story satisfying, you should generally give them an arc.
An arc is simply the transition of a character from point A to point C, with the events of the story shaping their transition from one to the other. Harry Potter starts out as a lonely, abused child living in the cupboard under the stairs, and ends up as a great wizard who saves lives and defeats evil. Luke Skywalker is a farm boy who becomes a Jedi. Dorian Grey is a beautiful, indolent hedonist who turns into a monster that cannot stand the sight of himself.
Arcs can affect physical change in the world and relate to a character's social status or personal deeds (like those of Harry and Luke) or focus purely around the psychological and emotional development of a character (Dorian). They can be either positive or negative changes, but the change must be meaningful and related clearly to the events of the story in a way that readers can understand. A character's arc is essentially a microcosm of a story in and of itself. Seeing someone change and react and develop in response to the events around them is in itself its own little personal story, independent of the larger plot.
However, it's important to keep in mind that "change" is not synonymous with "arc". Countless times I've heard readers, viewers, and players defend even the most absurd character transitions on the grounds that it's "better than no change at all", and I feel as though it's an assumption worth commenting on.
Character change is a lot like the concept of originality. We tend to hold it up as a prized quality in storytelling; something that every creator should strive for to enrich their work, and something that less experienced students of the art often trip up on in their efforts to recreate. Of course, this is silly. Originality has no intrinsic value in storytelling, because everyone can be wacky and weird and bizarre if they want to. In fact, the reason most stories come across as formulaic is because they're sticking to the same tried and tested set of rules that generally make for an entertaining story in the first place. Why would you try to be original when you're just going to be breaking something that already works perfectly fine?
It's hard to break what works and make it
still work after you're done toying around with all the parts, and that's where genuinely praiseworthy originality comes from. It requires skilful handling to be
good original rather than just
weird original, and the same is true of making character change into a satisfying arc rather than just change for the sake of change.
The worst instances of this can often be found in long-running TV dramas, when writers feel the need to switch up the character dynamics a few seasons down the line by changing who's a bad guy and who's a good guy (etc.) to keep things fresh. This usually runs into a whole bunch of pitfalls when the writers don't adequately explain or demonstrate the transition, making characters "change" purely for the sake of having something different happen.
I vividly recall the absurdity of the third season of the show
Heroes, where the good guy/bad guy dynamics started to switch up on an almost episodic basis, with half the cast coming off as schizophrenic at best and almost comical at worst. This is more of an issue with how these dramas are commissioned and written in general (if you give a character a complete arc in season one, what do you do with them in season two? If you leave their arc unresolved at the end of a season, what if the network cancels your show before you can finish it?) but it's a good reference point for some of the pitfalls of long-term character development.
Similar to how TV shows feature characters with complete arcs, followed by radical personality shifts (or backpeddling) to facilitate new arcs in season two, novel sequels run the same risk of forcing the writer to change their characters for the sake of making them interesting again if the original book was initially intended to be self-contained.
This is one of the reasons why my current WIP
Broken Moon is a loose sequel featuring new characters rather than a direct continuation of the preceding book, and any plans I have for subsequent sequels will likely continue in the same mold. I like to give characters complete, satisfying arcs in my stories, and I don't want to have to change up their personalities or invalidate any of their development for the sake of a sequel.
Of course, it's perfectly possible to continue arcs on after they apparently "conclude" (Luke Skywalker became the hero, but he's not a Jedi yet, and Harry became a wizard, but he hasn't defeated Voldemort once and for all), but it does require some forethought.
Phew, so those are my thoughts on character change for the day! Change is not synonymous with Arc, and it has no inherent value in and of itself. Character change should never be haphazard, and if you're going to be writing sequels, make sure you think about how to carry on that arc beyond the initial point of resolution without cooking up something contrived that's only going to alienate your audience.