You can keep your 1080p HDTV, my SNES games look a million times better on a monitor that's nearly 30 years old.
Classic video game consoles are best paired with classic displays. The last CRT TV I bought was a Sony FD Trinitron KV-21LS30U in 2001, and it got left behind several house moves ago. I’m still waiting for one to show up near me on eBay, but until then I’ve happened upon a surprisingly good alternative.
Enter the Acorn ADF-32 monitor that came with our BBC Master Compact system all the way back in 1986. I never realised it at the time, but it turns out this hits all the right buttons for 16-bit gaming:
My SNES is hooked up using a Nintendo RGB cable (which will also work with the same multi out socket on your GameCube or N64). It’s also fitted with one of the new SuperCIC mods so that it can play games from any region and switch between 50/60Hz modes on the fly.