Universal uniqueness criterion?

[bo198214]

*Fortunately the derivatives at the fixed points L and L* is not a real number, (they are 0.318...+1.337...i) Of course I'm talking about the fixed points of the principal branch of the logarithm.

Yes, that should be likely for fixed point not on the real axis.

*What is the problem with the extension overlapping with the original domain?

Thats perhaps not so much the problem, more of a problem is that it is no domain anymore - boundary crosses itself.

*What initial regions do all of the known methods of complex extensions of tetrations have?

The property of a region to be initial does not depend on the super-function, but just on the function itself. So every initial region of exp can possibly be a domain of definition for tetration. I think/hope every method returns values on the default initial region - the crescent - of all z such that |z|<|L| and such that Re(z)>Re(L). But of course it may still return values on other initial regions.

And still it could be possible (though I dont think so and am close to prove my conjecture) that one tetration is biholomorphic only on one initial region and the other tetration is biholomorphic only on the other initial region.

For a picture of iterated exponentiation have a look at figure 8 in Dmitriis article
http://www.ams.org/mcom/2009-78-267/S002.../home.html
The black lines are the integer iterations of the straight line from L to L*.