in fact, due to how youtube compression work, watching 4k does resulted in higher bit-rate stream and better image even with 1080p monitor.
If you want to watch YouTube above 1080p is your GPU decoding support. Below 1080p, YouTube can use either VP9 or H.264. Above 1080p, YouTube only use VP9 (well, AV1 also used but it still pretty early in adoption). H.264 hardware decoding is supported by a lot of GPU. VP9 hardware decoding only supported by ryzen based APU, Navi GPU on AMD side. On Nvidia, GTX 950, 960, 750 GM206, and Pascal and newer GPU. Intel iGPU only from Kaby Lake and newer. Though, Intel iGPU from my testing only able on handling 1080p60 max. For system with other GPU, decoding would be handled either purely by CPU or by the Hybrid mode (using shader instead of a dedicated silicon).
By default, YouTube always use VP9. To force YouTube to use H.264, install h264ify extension(available on Chrome and Firefox).