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