I think it is ; Ocimum kilimandscharicum - Camphor Basil .

Do all members of the mint family have square stems? If so, does this plant have a square stem? ...you have to touch the stem to be sure of "squareness".
Also, do any basils have square stems?

I think it's yes to both inquiries, probably all basils, basils are mints (I believe).

Here a photo of me " Cheating the seasons"
A few outdoors , this I brought in , it was laying outside on top of the " dirt " with roots , and sprouts ..
I am hoping for more seeds , so an earlier start in a propagation container ..
Round Stem ,,
Roots new from outside ,, That is really hardy for a Basil

It sure is acting like my Chocolate Mint does. Mine changes colors in winter, that is only the parts of the plants that are not in a protected are. It has the same runners that mine have. I have have my plant for ten years. It of course has grow through the planning bed where I have let it. My orange mint did all the same things except die off for winter. Both my mints have different shaped stems depending on which part of the plant you feel. I don't really know what it is but the pictures in the summer should definitely give you a solid answer!