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 finished the game.
You raised Cecil (Paladin) to level 70.
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 raised Cecil (Paladin) to level 50.
You earned 200,000 gil.
You defeated 1,000 enemies.
You completed 60% of the Bestiary.
You defeated Dr. Lugae.
You defeated Calcabrina.
You earned 50,000 gil.
You defeated Barbariccia.
You defeated the Magus Sisters.
You raised Cecil (Paladin) to level 30.
You defeated Dark Elf.
You completed 30% of the Bestiary.
You raised Cecil (Paladin) to level 20.
You defeated Cagnazzo.
You defeated 500 enemies.
You talked to everybody in the Training Classroom.
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 obtained one augment.
You acquired one summon.
You raised Cecil (Dark Knight) to level 15.
You raised Cecil (Dark Knight) to level 20.
You raised Cecil (Paladin) to level 10.
You earned 10,000 gil.
You defeated 100 enemies.
You completed 10% of the Bestiary.
You played a new 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 20 augments.
You obtained all augments.
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 25.
You earned 500,000 gil.
You defeated 2,000 enemies.
You completed the entire Bestiary.
You defeated Geryon.
You defeated Proto-Babil.
You talked to everybody in the Developers' Office.
You earned all achievements.