I love the Fantasy Zone games I've played thus far. I have the original Fantasy Zone on XBox 360 (Arcade emulation), 3DS (3D Fantasy Zone), Turbografx and Master System. Unlike most people who prefer arcade perfection, the SMS is definitely my favourite - it feels fluid, has better screen scrolling and a fairer difficulty than the Arcade game.
I've only played the 3D Classics version of Fantasy Zone II which is actually a completely different game to the original version. The original was developed for Master System before being ported to arcades. M2 then developed a special version of the game for the PS2 Sega Ages compilation of Fantasy Zone which rebuilt the entire game using 80s style Arcade graphics like the original Fantasy Zone, as well as adding loads of new mechanics and levels and stuff. That's the version they used for 3D Fantasy Zone 2 on 3DS.
I need to pick up Super Fantasy Zone on Mega Drive and the original version of Fantasy Zone 2 on SMS sometime to finish the set. I can live without Fantasy Zone: The Maze though
EDIT: Also, maybe this is why you thought the series was like Space Harrier - there's a cancelled game that was in development once upon a time called Space Fantasy Zone, which is literally cutesy Space Harrier with Opa-Opa.
