No. When you buy Final Draft, you get a license for both platforms. You can download the full Win and Mac installers for Final Draft HERE.
The license allows you to run Final Draft simultaneously on up to two computers that you own and operate, and it doesn't matter whether it's one Windows and one Mac computer or two of either.
If you're running FD on two computers and you want to run it on a third, sign out of one them (FD13) or deactivate one of them (FD12 / 11 / 10) so the seat will become temporarily or permanently available for that third computer. Note that the computers must be connected to the internet when you sign in or out (FD13) or activate and deactivate (FD12 / 11 / 10).
Also see If I purchase Final Draft from the Mac App Store, can I install it on my Windows computer as well?
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