How to get the Bahamut Summon in FF4
The Cave of Bahamut is found on the moon almost directly east of the Crystal Palace (the huge crystal tower in the center of the moon). Finding this cave shouldn't be too difficult considering it stands out on the moon's relatively small surface.
Note: In order to get the Bahamut Summon you'll first need to complete the Asura and Leviathan Summon side quests in the Land of Summons. If you've obtained both of them, you'll be able to get Bahamut.
Due to the difficult of Bahamut it's recommended that you first defeat the Crystal Palace and obtain FuSoYa as a party member. This will make both the Cave of Bahamut dungeon and the fight with Bahamut much easier.
Cave of Bahamut B1 Treasure Chests:
Genji Shield (Hidden passage in the northern part of the rock)
Cave of Bahamut B2 Treasure Chests:
On the Cave of Bahamut B2 when you're crossing the narrow bridge you'll encounter your first Behemoth enemy. There are two more of these that you will be fighting before the battle with Bahamut and you should treat them like their own boss battle.
Behemoths hit extremely hard but primarily only attack when you attack them. They are also immune to the majority of spells, the only spells that hit them are Flare and Holy, two spells FuSoYa has. Despite these spells hitting the Behemoth, it'll cause them to counter with the Maelstrom ability which will drop your entire party's health to almost 100.
To defeat the Behemoth enemies you'll want to use the spell Blink which both Rosa and FuSoYa will have. Edge's Image spell will work too but it is a self buff only. This spell will allow you to dodge the majority of the Behemoth's counter attacks which makes the entire fight significantly easier. Once everyone is buffed with Blink unload on the Behemoth with Cecil and Edge. I'd recommend defending each round on the others.
You'll find Bahamut on B3 of the Cave of Bahamut and en route to him you'll bump into two more Behemoths, use the same strategy to defeat all three of them. If you really need to, leave the dungeon and use a Cottage before returning to defeat Bahamut.
Speak with the human on the top of the pyramid (shown in the screen shot above) to begin the battle with Bahamut. The strategy for this fight is pretty simple, if you have FuSoYa use him and Rosa to cast Reflect on every single party member. Reflect is the only way that you're able to survive the Mega Flare ability that Bahamut uses.
When the fight begins you will see a timer at the top of the screen that says "5". It will count down throughout the fight until Bahamut uses his Mega Flare ability. Your job is to get Reflect on as many people as possible before Mega Flare while also dealing damage to Bahamut with Cecil, Rydia and Edge.
Bahamut is not weak to any magic in particular so hit him with the strongest -aga or -aja spells that you have. If anyone dies use Life on them and then Reflect immediately. There is no other damage during this fight other than Bahamut's Mega Flare.
In total Bahamut has 45,001 HP, each time you Reflect his Mega Flare you deal 9999 damage to him which will speed the fight up a lot too. Once the fight is over, as always, Rydia is taught the summon Bahamut. You can now Teleport out of the dungeon ot use an Emergency Exit.