What genre would it be?
I'd start with a 3D platformer, similar to Crash Bandicoot 1.
What would it be called?
If I'm feeling ballsy, I'd just buy the rights to Crash Bandicoot from Microsoft and make a full-fledged sequel called Crash Bandicoot: Paradox. Simple, yet effective. However, considering that'd be nigh impossible even with a successful business, I'd probably just make something wacky and silly, similar to Rayman.
What would it be about?
Assuming it'd be Crash, the plot would take place after the perfect ending to Warped; the time twister goes haywire and starts blistering the universe with portals leading to various locations throughout time and space, bringing every gem and crystal with them. In order to stabilize it and destroy it properly, Aku Aku sends out the entire marsupial family to gather them. However, as they set out on their journey, Cortex has not forgotten his hatred for Crash as a baby, setting out on his own quest to retrieve the machine and regrasp his grip on the universe.
How does it play?
Seeing that most 3D platformers tend to be laid-back and collectathon-y, this one would be based on stages where the objective is simply to get to the end. These stages would be laid out in a hub world like CTR/NK, with several zones each with its own theme, and every zone has 2 mini-bosses and a big boss. Mini-bosses give you crystals which you need to progress to other zones, while big bosses unlock power-ups that make you more powerful and let you be able to access secret areas.
Like previous games, each stage has a gem upon getting all the boxes. Gems unlock secret doors that lead to bonus treasure, character unlocks, or even challenge levels.
Also implemented are new masks. Aku Aku would, of course, me temporary invincibility. Other masks include one that makes you grow/shrink, another would give you the ability to dive underwater or walk on water, etc. The idea is that you unlock masks to access certain paths and areas in levels that you couldn't during a normal run-through, similar to skull routes.
Also a shit-ton of easter eggs.
Who would you have publish it?
It's hard to say, considering most companies tend to have at least some shade in how they do business. Also it really depends. If anyone's willing to make it as accessible as possible, I'd be down with that, though I'd want to give credit where it's due and have Sony do it, since it's essentially their IP and I don't care what y'all say. And hell, make it a year exclusive to sell consoles, cause why not?
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Curious to see what others say.