I’m half way writing a book on the Mac and need to change to a PC

I’m half way writing a book on the Mac. ¿Can I finish it on a Windows based computer?, ¿Will I have to purchase a new software?

Yes, and yes.

Are you using Mac Scrivener 3? The Scrivener -> About Scrivener command will tell you.

If so, then BEFORE you dispose of the Mac, use the File -> Export -> As Scrivener 2 Project command to revert to the older project format that the release version of Windows Scrivener uses.

I’d also recommend making sure that everything is up and running on the PC before you let the Mac go.