Here is the full list of all 58 Final Fantasy IV (3D Remake) achievements. It takes around 80-100 hours to unlock all of the achievements on Windows.
You talked to everybody in the Training Classroom.
You played a new game.
You traveled to Mist.
You traveled to Damcyan Castle.
You defeated the Sahagin that attacked at Kaipo.
You defeated Mom Bomb.
You were shipwrecked by Leviathan.
You defeated Scarmiglione.
You defeated Cagnazzo.
You defeated Dark Elf.
You defeated the Magus Sisters.
You defeated Barbariccia.
You defeated Calcabrina.
You defeated Dr. Lugae.
You defeated Rubicante.
You defeated Demon Wall.
You went to the moon on The Lunar Whale.
You defeated the elemental archfiends.
You defeated the CPU.
You finished the game.
You finished the game on Hard difficulty.
You finished the game twice.
You obtained Excalibur.
You obtained Ragnarok, Holy Lance, Murasame, and Ribbon.
You obtained Adamant Armor.
You obtained all the Onion items.
You obtained one augment.
You obtained 20 augments.
You obtained all augments.
You acquired one summon.
You acquired all hidden summons.
You acquired all summons.
You traveled across the entire map.
You played through all Namingway events.
You raised Cecil (Dark Knight) to level 15.
You raised Cecil (Dark Knight) to level 20.
You raised Cecil (Dark Knight) to level 25.
You raised Cecil (Paladin) to level 10.
You raised Cecil (Paladin) to level 20.
You raised Cecil (Paladin) to level 30.
You raised Cecil (Paladin) to level 50.
You raised Cecil (Paladin) to level 70.
You earned 10,000 gil.
You earned 50,000 gil.
You earned 200,000 gil.
You earned 500,000 gil.
You defeated 100 enemies.
You defeated 500 enemies.
You defeated 1,000 enemies.
You defeated 2,000 enemies.
You completed 10% of the Bestiary.
You completed 30% of the Bestiary.
You completed 60% of the Bestiary.
You completed the entire Bestiary.
You defeated Geryon.
You defeated Proto-Babil.
You talked to everybody in the Developers' Office.
You earned all achievements.