User blog comment:ArtismScrub/Proof that negative tetration is impossible/@comment-32213734-20180323013202

This is about integer negatives, but what about real negatives? Maybe tetration behaves like Gamma function? We may define fractional tetration, for example:

a ↑↑ (b + 0.5) = f(a ↑↑ b), where f(x) is functional square root of ax: f(f(x)) = ax.

If a ↑↑ b < 0 for -2 < b < -1 then a ↑↑ (b - 1) = loga(a ↑↑ b), that is logarithm of negative number, but complex logarithm has infinite many values, so, it is possible that tetration also has infinite many values.

But I'm not sure.