You appear to be trying to move a running program from one machine to another. Not generally possible, and certainly not on any Windows based system that I know of.
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You can't (generally) use your laptop screen as a second screen for your desktop, though it's quite likely you could use your monitor as an additional screen from your laptop.
Short version - buy a bigger screen (or better two) for your desktop computer. You will save more time and effort (ergo: money for your employers) in the space of a few months than trying to hack around with multiple computers to do what could be trivially accomplished with a decent gfx card and a couple of $100 screens.