The only other peripheral you'll need is the GameCube controller USB adapter Nintendo released for use with Super Smash Bros. on the Wii U. It's still possible to pick up both the official and third-party versions of this USB adapter.
Regardless of the reasoning, GameCube controllers now work on the Nintendo Switch. As the name implies, this feature updates the game to all users on their latest versions, which is useful for users who have multiple profiles on one system. Today developer Terry Cavanagh and publisher Nicalis, Inc. announced that the popular platforming game will be making its way to Nintendo's new console on November 17, 2017. In our newest video we give 1 vs. 1 (COM), 1 vs. 1 and 2 vs. 2 a try while juggling a few GameCube controllers. After plugging it all in, the GameCube controller will identified as a USB pad in the Switch's controller menu.
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NintendoWorldReport has also confirmed that WaveBird controllers also work.
Videos going around show that GameCube support isn't fully baked.
This is where things get a little hairy, though. If you're looking to experience Super Mario Sunshine nostalgia whilst playing the soon to be released Odyssey, this should be music to your ears. The Gamecube controller lacks a left bumper button, a minus button, home button, and motion support. Honestly, I think the second is more likely, as Nintendo is getting ready to launch their paid online and will likely have Virtual Console ready for that. As it turns out, the Nintendo Switch now supports GameCube controllers.