There is a huge problem with this quest often not being marked as complete after you finish it. The short answer is it often won't complete if there are other people doing it ahead of you and you should just try again until it does.
Now the long answer: There is a ton of misinformation here about what you need to do to get the quest to complete. There is not really a "bug" in the quest and taking off buffs, dismissing pets, logging in and out or moving a certain way after landing will not help in the slightest. It's basically just luck so when it works people assume that whatever they did must have fixed the problem.
Here's what's going on:
When you start the quest you fire a flare and Keristrasza appears and runs through her script. This takes around 90 seconds. As everyone quickly figures out, no one else can summon Keristrasza until the previous person's Keristrasza has despawned. The instant Keristrasza despawns and is replaced by a mount that takes the first quest-giver away the next person can summon her. So if you are trying to win the flare contest with a long line of questers, fire the instant she fades out to be replaced by the mount version.
As other people have correctly noted, you will not continue with the mount part of the quest if you cannot mount at that time (shapeshifted or already mounted or in combat).
The second part of the quest involves flying around and looking at things.
The third part of the quest works just like the first part but it's much less obvious because you're not directly controlling it. What happens is you land and then Keristrasza appears, tells you to run and then gets captured by Malygos. This seems to take a variable amount of time-sometimes it happens right away, sometimes it seems to take a few minutes. The important part is that, just like at the start, when you land the Keristrasza that appears is "your" Keristrasza and only one Keristrasza can be spawned at a time. The issue therefore is if you land and someone else's Keristrasza is still going through her script (even with just a second left), your Keristrasza will fail to spawn and you won't complete the quest. What happens is one person starts the script and the next two people in line (who are running through the first part as fast as possible) fail to complete the quest while the first person's script runs. Repeat.
As you can see, whether or not you complete the quest is entirely due to luck-whether or not the previous person's Keristrasza is still up. If she is, you just have to try again.
Due to the apparent variability in the time it takes the final script to run I don't think it's that useful to try to time your run but it may be possible. You'd need a friend over there telling you what was going on…
It's not really a bug so much as bad design. If you were to do the quest when no one else had done it in the past 5 minutes it would complete every time. That's why this wasn't discovered in Beta and the issue will largely go away when the initial rush of the expansion is over but you'll still see it when you've got a group going through the quests together.