Bubble Bobble is in my top 5, and I have a ton of them. I used to play all kinds of ports (C64, Atari, NES) but I eventually got the Amiga version as my first original game (I started out with a bunch of copied floppies). It was messy as you had to disable the 1MB memory expansion to get it to work.
Actually, from what I've heard in a feature in Retro Gamer, all the computer ports were made pretty much by playing the arcade and recreating it, meaning that things the creator couldn't find didn't make it in. And since the game has a TON of obscure rules to decide drops and powerups, no version is the same. Apparently the NES version was the only one that Taito did themselves (Sega ported the game for themselves since Taito weren't allowed by Nintendo to be a third party).
The NES version was great for adding the 20 extra stages and having password feature, but it wasn't perfect.
I also played Rainbow Islands a bunch (it was included with my Bubble Bobble) but never got too far in it. Also had Parasol Stars on Amiga and didn't know how uncommon that was.
I remember picking up the collection for PSone but didn't really enjoy it, I think it was also a weird port.
In 2001, I picked up a Saturn with Bubble Symphony. It was a decent game overall but a bit messy and gaudy. Parasol Stars actually works better as a sequel. I keep forgetting to have a look at Bubble Memories.
Bubble Bobble: Old & New was notable for being the first "complete" portable Bubble Bobble, and it was awesome. I did have to buy it twice, as the first copy turned out to be a bootleg (thanks, Ebay).
Bubble Bobble Plus / Neo was okay but once again there was something faintly off with the controls. It's actually a recurring thing. Bubble Bobble is EXTREMELY detailed with its gameplay, and every single change can be devastating. How bubbles move around the stage, how easily they break, what part of your character actually breaks the bubble, what platforms you can jump through, the exact collision detection of each platform, etc etc etc. This makes some stages nearly impossible in the ports.
Which brings me back to the arcade version. I finally got an arcade cabinet (for Virtua Fighter 3), then rewired it for Jamma, then got an original Bubble Bobble PCB with an adapter (since Bubble Bobble itself was pre-Jamma), so I can play the original. And it IS the best version by far.
And finally, the Arcade Archives version for PS4. It seems to me to be 100% arcade perfect, at least everything feels that way after a few playthroughs. Since getting a cab and a $200+ PCB is out of the question for most people, that's the version I would recommend.
(Side note: these plushies were released by Gaya Entertainment a few years ago, and they are AWESOME!)