Universal uniqueness criterion?

[Kouznetsov]

.. Well, but \( \exp \) *has* more fixed points. In every strip \( 2\pi i k < \Im(z) < 2\pi i (k+1) \) there is a fixed point of \( \exp \).

Henryk, how about to build up "another" holomorphic tetration that goes to other fixed points at \( \pm i \infty \)?
And what will do such a function at \( x+iy \) while \( x \rightarrow -\infty \) ?