HDMI screen switch

On the computer you just plug it into the HDMI port

If you connect it to the HDMI in on the Xbox One, you will experience lag. This is meant for TV

You should just get a switchbox or plug it into another HDMI on the TV if this is your objective.

It really depends on the TV you are using. If you have a 1080p screen, definately go with a HDMI. I saw a difference switching between a HDMI cable and component cables. Any other screen resolution and I saw little to no change.

Depending on distance you could hook up a HDMI cable from the Xbox One to the TV. To make it even easier you could buy a HDMI switch and plug the console into that with one cable running to the computer monitor and another to the big screen. Then you could
switch with a single button press. Would buy a switch that is HDCP High Definition Content Protection compliant.

Image quality could be effected if things like resolution and color settings need to be changed when switching between screens though.

hdmi switch is not the way to go, but a hdmi splitter that then splits the source hdmi to 2 outputs, i use one to my monitor and to the big screen, basically clones the image so the resolution has to be set accordingly.

my monitor is 1080p and tv 4k, so when i switch the monitor on i change the xbox res on the tv to what the monitor is and then carry on playing.

You need to look for a "HDMI Switch" with 1 input and 2 outputs. This will then give you a phyiscal switch which you can toggle at will to get the screen displayed as you wish without unplugging.