PuPu the Alien Side Quest - FF8 Guide
You can begin the PuPu Side Quest any time after completing Lunar Base on Disc 3. For this quest you'll want to use Enc-None since you're going to have to run around and get into encounters with PuPu. Enc-None will prevent all other encounters except PuPu.
For this quest you will need to visit four different locations, the area around Winhill, Mandy Beach, Heath Peninsula and Kashkabald Desert. All four of these locations are marked on my screenshot below. At each location you'll want to run around until you encounter PuPu. You don't have to do anything in the battle, just wait until it ends. An example of one of these battles is shown in the screenshot above.
If you're struggling to encounter PuPu at any of these locations use my screenshots below the map as a guide. Only two of the locations are actually difficult to find which is why there's only two screenshots for the four different locations.
Once you've visited all four locations you'll want to go to the rocky area surrounding the northern portion of the Grandidi Forest (pictured below). Land the Ragnarok up here and run around until you encounter the flying saucer - this time you're actually able to attack it.
A few hits will bring it down and your reward for completing the battle is an Aegis Amulet. The Aegis Amulet item will teach a GF the Eva-J ability which is extremely useful since you won't acquire 3 GFs with this ability in the game. That means the only way to Junction magic to Evasion on all 3 characters in your active party is to use this item.
Last but not least there is one final part to the PuPu quest which involves encountering PuPu himself where Balamb Garden used to be. When we gained control of Balamb Garden during Disc 2 it came out of the ground and left a big hole, return to that location near Balamb Town and run around the edges of that hole.
It'll only take a few steps until you encounter PuPu the alien. During this battle you have two options, you can either listen to PuPu and give him 5 Elixirs which will reward you with the PuPu Card item. This is a one of a kind item and it's strongly recommended that you feed him the Elixirs for it.
Alternatively you can punch PuPu in the face and get the Accelerator item as a drop. You could also Devour him for Speed +1 but that's basically a waste too since there are many other easier ways to boost your speed stat, all of which are talked about in my Stat Item Farming Guide.