Here is the full list of all 51 Crash Bandicoot 4: It's About Time achievements worth 1,000 gamerscore.
Watch the 106% bonus ending
Earn ALL of the platinum Time Trial Relics
Earn ALL N.Sanely Perfect Relics
Play a game of Crate Combo
Make music in an unlikely place
Play a game of Checkpoint Race
Complete a level with Pass N. Play enabled
Watch the 100% bonus ending
Earn ALL N. Verted Gems
Complete ALL levels in N. Verted mode
Earn ALL of the platinum Flashback Relics
Aim and fire!
Earn ALL of the Clear Gems
Find the hidden Blue Gem
Make some noise!
Find the hidden Red Gem
Beat any boss in N. Verted mode
Complete all Timelines
Find the hidden Yellow Gem
Earn any N.Sanely Perfect Relic
Complete a level as Coco
Earn any Time Trial Relic
Defeat an enemy with Triple Spin
Pull off a Triple Spin
Defeat Dr. Neo Cortex... again.
Betrayed! By your worst enemy.
Defeat the Doctors N. Tropy
Grabbed a bushel of Bumpa
Earn half of the Clear Gems
Meet Ika-Ika
Defeat Dr. Neo Cortex
Meet Kupuna-Wa
Do some target practice
Scare the cat hiding from a party
Beat a level in N. Verted mode
Defeat N. Brio
Find the source of the Bumpa Berries
Shut N. Brio up
Complete any Timeline
Beat a story level without dying
Meet 'Akano
Earn any Flashback Relic
Try on a Skin
Defeat N. Gin and his Weapon of Mass Percussion
Find the hidden Green Gem
Acquire a Flashback Tape
Shut N. Gin up
Meet Lani-Loli
Complete a Bonus Path
Crouch or slide into a double jump
Spend some time surfin' the waves