In my experience, 90% of the time alternate endings are weird. I do not like them in traditional storytelling, and I feel like the majority of stories that incorporate more than one ending do so inappropriately. Alternate endings do have their place, but they require a very specific type of story in order to pull it off. I'll be mentioning video games again here, because recently the gaming medium has been the only one to tackle alternate endings in a widespread manner (to varying degrees of success).
I'm going to be bold and outright say that no traditional story should ever be given an alternate ending. Yeah, you can argue that they're an interesting side feature to see on a blu-ray, much like deleted scenes — but that's all they are. An interesting extra. When you have movies like 28 Days Later that screened with the alternate ending being played after the first one, it creates a jarring lack of closure for the audience. The fact that the directors preferred this alternate ending to the one in the final cut (and only changed it because it tested badly with audiences) further muddies the water. I have no problem with people changing the endings to their stories pre-release for whatever reason, but I feel as though offering two very different conclusions side by side undermines the integrity of a story, and damages the product as a whole. It becomes hard to discuss a story critically when you have to view the entire thing in the context of not one, but two (or more) separate conclusions. Ultimately, it's confusing.
Now that isn't to say that viewing the same material through different filters can't be very interesting and provocative — but it all depends on the type of story you're trying to tell. As I mentioned, traditional storytelling relies on a single, conclusive, wrap-everything-up-and-give-us-closure type of ending. So what's an example of a good alternate ending? The recent Deus Ex game, Human Revolution offers the player a choice of no less than four different endings. Surely that's even more awful and convoluted, right?
Well, no. Because from the start, Human Revolution is set up as a commentary on the theoretical (for now) issue of transhumanism becoming a central part of society. Throughout the game the player is presented with a series of moral viewpoints that explore different sides of the issue, and the entire storyline is deliberately ambiguous on which one of these viewpoints might be considered the "right" one. As a result, any conclusion that favoured one perspective over the others would undermine the entire debate the game intended to provoke. It feeds the player information about the issue throughout the entire storyline, and at the end presents us with the question: "What do you think? What conclusions have you reached after all of this?"
As a result, each ending provides a different monologue that concludes the protagonist's story with him reaching a decision (or not, as one option allows) about his stance on transhumanism. Without alternate endings, the story could never have provided as satisfying a conclusion on the subject without becoming biased in a viewpoint that the player might not agree with.
Unfortunately though, Human Revolution is the exception, not the rule. A lot of games handle alternate endings awfully (at least from a story perspective), often concluding in a "bad ending" if the player hasn't jumped through the requisite hoops to see the "good" or "true" ending along the way. On a game design level this ties into things like replayability, but story-wise it often ruins the conclusion to a narrative by purposefully giving the player an unsatisfying ending to their experience. It's even worse when story elements are deliberately left unresolved in these "bad" endings. That's straight up crippling your storytelling on purpose.
So, while I do think alternate endings have their place in a certain kind of narrative (namely, an ambiguous one), they really add nothing to a traditional narrative. In fact, I'd argue that they damage it by feeding the audience conflicting information.
And let's not even get into going back to change your ending post-release because people didn't like it.