Final Fantasy IX is unique compared to other games in the Final Fantasy series in that there are no explicit “Ultimate Weapons” for all the characters. Below is a list of what are arguably the best weapons for each of the characters and where to find them:
Zidane
The most powerful weapon for Zidane is the 
Ultima Weapon. You can find it by using a 
Dead Pepper near the fourth Dive Spot near the Shimmering Island, which is part of the Chocobo Dive Spot side quest. You can only reach the fourth and final dive spot during Disc 4 of the game.
Ultima Weapon has 100 attack power, teaches the Flee ability, and can inflict the Sleep status effect.
Vivi
The most powerful weapon for Vivi is called the 
Mace of Zeus. . Memoria in the “Gaia’s Birth” room. The Mace is hidden on the left side of the platform, partway up the ladders.
Mace of Zeus teaches Vivi his most powerful spell: Doomsday. It can also inflict the Mini status effect.
Steiner
Steiner has two Ultimate Weapons, although one of them is extremely difficult to get. The 
Excalibur II is not only the most powerful weapon for Steiner, it is the most powerful weapon in the game.
The weapon can only be obtained by completing the Excalibur II side quest, which involves reaching the end of Memoria in under 12 hours—arguably one of the most difficult side quests ever created amongst all Final Fantasy titles.
Excalibur II teaches Steiner the Minus Strike, Climhazzard, and Stock Break abilities and has 108 attack power.
Because of the difficulty of obtaining Excalibur II (which you basically have to spend the full game speeding through just to obtain it) the second-best weapon for Steiner is normally considered is Ultimate Weapon. It is the 
Ragnarok, and you can obtain it by locating Chocograph #21: Outer Island through the Chocograph side quest.
Ragnarok teaches Steiner the 
Shock and Thunder Slash abilities.
Dagger
The strongest weapon for Dagger is called the 
Whale Whisker. You can obtain it by locating Chocograph #18: Ocean, which is part of the Chocograph side quest.
Whale Whisker can teach Dagger the Curaga and Life spells.
Freya
Freya’s Ultimate Weapon is called the 
Dragon’s Hair lance. You can obtain it by opening the third Chocobo Mountain Crack using a Dead Pepper while standing in front of it. Check out the Chocobo Mountain Crack guide in the Side Quests section for more information.
Dragon’s Hair can teach Freya the Dragon Breath abilities, which deals a good amount of damage to enemies (Target’s max HP minus the current HP, thereby doing more damage later on in a fight).
Quina
The 
Gastro Fork, Quina’s strongest weapon, can be found by completing the Catching Frogs side quest. Check out the Catching Frogs guide in the Side Quests section for more information.
The Gastro Fork can teach Quina the High Tide ability and can inflict the Stop status effect.
Eiko
Eiko’s Ultimate Weapon is called the 
Angel Flute. You can find it in Memoria in the “Past” room (which is the one that has the Save Point up on the pedestal). There is no treasure chest—the Flute can be found by searching the area and pressing the 
or 
button repeatedly.
The Angel Flute teaches Eiko all the strongest healing spells, specifically Holy, Esuna, and Curaga.
Amarant
The 
Rune Claws are Amarant’s Ultimate Weapon. They can be found in Memoria as well in the “Time Interval” room, which is the one with the stairs leading up to the giant eye (the Invincible ship). Search the area off to the left of the stairs to find them.
The Rune Claws will teach Amarant the “Spare Change,” “Demi Shock,” and “Revive” abilities. It can also inflict the Darkness status effect.