Nope, connecting a high-resolution monitor should have no performance impact on the laptop. While gaming, the resolution the game is rendering at will make a big difference to frame rate, but not the resolution of the display itself.
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I run two 4K monitors, so that's 2x 3840x2160 which is way more pixels and it's totally fine.
Which monitor and which connection port do you use?
No, if you were doing some 3D application like video games or CAD to a lesser extent, it would put load on the graphics side, but the computer wouldn't 'slow down'. Real note on games, this is a really high resolution, I'd avoid all but the lightest games at this native resolution
Have 5120x1440 and a 3080. Will barely pull 30 FPS in heavy AAA games at native (Thanks DLSS), but just watching two full screen videos at the same time no issues at 120 hz. As long as it isn’t 3d or ai workloads you should be fine!