Here is the full list of all 16 Alex Kidd in Miracle World DX achievements.
Defeat Ox
Defeat Grizzly Bear and Funky Monkey
Defeat Gooseka in the Radaxian Castle
Defeat Chokkina in The City of Radaxian
Defeat Janken The Great
Discover the secret island
Beat peticopter level without shooting.
Obtain the Telepathy Ball
Free Egle
Speak with everyone in the game
Beat the game without buying any item in shops
Pet the sad doggy
Get all collectibles
Complete the main story
Beat the Jankenpon Tournament
Defeat Parplin in Nibana Lowlands