Here is the full list of all 21 Tempest 4000 achievements.
Obtain all trophies
Reach level 51 in Survival mode
Survive to level 25 in Pure Mode with no loss of life
Finish all game levels
Reach level 51 in Pure mode
Reach level 51
Survive to level 17 in Pure Mode without loss of life
Max out lives counter
Complete level 1-33 without dying
Get to level 25 in Pure Mode without jumping
Get to level 17 in Pure Mode without jumping.
Get to level 16
Get 5 or more hearts during warpout.
Finish one Path Bonus round
Enter both Bonus Round types in one game.
Get to level 9 in Pure Mode without jumping
Survive to level 9 in Pure Mode without loss of life
Finish one Rings Bonus round
Get the ANGEL bonus powerup
Get rescued by AI droid at distance
You and the AI Droid both get 25 or more kills on a level