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