My company gives everyone a laptop and an external monitor. Normally you plug the laptop in to the external monitor for extra screen space, but if you want to roam around the office you unplug it.

I set this sort of thing up with my laptop. The laptop sits on a high shelf next to the sitting desk where it can serve as a standing desk monitor. It's attached to an external monitor on the sitting desk. If I want to use a standing desk configuration, I activate the laptop display and move the keyboard and mouse to a shelf below the laptop. I could make it even more convenient and have two keyboards and mice constantly attached to the laptop, one pair on the standing desk shelf and another on the sitting desk.

Haven't used the standing desk position much though.

Either put the monitor somewhere where it can easily be used from either position, or, if this isn't feasible, have a second monitor that displays the same thing as the first.

It seems like it would be less of an inconvenience, since you could switch between them without even having to save and load files.

You need to start the other computer and load the programs that you need. I doubt that's done in 3 seconds.

There are devices in which you can plug an additional computer, mouse and keyboard and switch which set is connected to a given computer.

Yes, I have two monitors on each. This cost isn't that large in part because each computer has one high quality monitor and one cheap one.