Real and complex tetration

[Gottfried]

(...)  But complex valued tetration would be what your tetration would be called. And real valued tetration would be a tetration that preserves real values.

A real valued tetration can still produce complex numbers, but only for complex arguments.

A complex valued tetration can produce complex numbers for real arguments.
(...)

Hi buddie ( ;-) ) -

 well said.  

 My most liked example is that of the fractional interpolation of the fibonacci-numbers.   

 - Do we want to be real to real?
 - Do we want to be real to complex?

And then: why?

The -for me- most natural interpolation is that along the Binet-formula. But this gives real-to-complex interpolation. 
In wikipedia one finds the real-to-real interpretation, which is in my view an ugly hoax, siimply introducing the \( \cos() \)-function to distort the real-to-complex way. 
(a small discussion as a result of a random observation in a discussion of fractional iteration of \( f(x)={1 \over 1+x} \), see plot-example in section 5)

Over the long run, I think, the decision shall be made once we observe a fibonacci-process in nature and have as well reason to assume that the interpolated process is in nature as well. Before that might occur, I think we can only decide in terms of (mathematical) elegance, versatility, ... .

Well, this is a toy model, but it might sharpen the mind for the argumentation when we insert our problem "tetration" instead the term "fibonacci" and ask for real-to-complex or real-to-real-interpolation method ...

Gottfried