What are all the tetration extension methods?

Question

Tetration is the next step in our regular operations. Addition, multiplication, exponentiation, tetration.

It is constructed by repetitive exponentiations. "$a$ tetration $b$" is written $^{b}a$

For example, $^{4}2=2^{(2^{(2^2)})}=65 536$. With this definition, it is only defined for integer hights greater than $-2$.

($^{-1}a=0, ~^{0}a=1$)

There are no current accepted definition of tetration with non integer hights. (Or complex). For example, $^{1.5}2$ isn't defined, nor is $^i2$

I've done a lot of extensive ressearch on the internet and found many things, many reccurent authors and different methods. They are scattered, and really hard to understand (at least for me)

Such as this calculator : http://myweb.astate.edu/wpaulsen/tetcalc/tetcalc.html#:~:text=This%20calculator%20finds%20the%20tetration,(z
And other websites : https://tetration.org/index.php/Main_Page https://en.citizendium.org/wiki/tetration
Tetration forum, etc

I would like to put order into all that scattered work into the internet. But I usually don't understand these methods nor know all the details, properties and implications of these. I would need specialists in the field, or at least people with more knowledge than me.

The thing, would be to make a list of all these methods, with :

1. Name of the authors/theorems used, etc
2. A simple explanation of the basis of the method
3. If it is continuous on $^xa$ and $^ax$
4. The properties that are satisfied
5. The domain of definition (hight and base)
6. If some of these methods give exactly the same results

For the continuity, it is normal to find discontinuities in the negatives, since there are veryical asymptotes for all integer negatives smaller than $-1$. ($^xa$)

And for the properties, here's a list. Let's write $f$ the function $f(x)=a^x$. In that case, $^xa=f^x(1)$.

1. $^{x+1}a=f(^xa)$
2. $f^n(f^k(X_0))=f^{n+k}(X_0)$

I have my own method, which is one of many. If I had to present my method by the "rules" of that post, it would be :

1. Koenig's solution to Schroeder's equation.

2. It uses attractive fixed points. $f(x)=a^x$. It solves the equation $Ψ(f(x))-τ=λ(Ψ(x)-τ)$ Where $τ$ is the attractive fixed point of $f$ (so where the infinite tetration of $a$ converges), $λ$ a constant equal to $f'(τ)$, and $Ψ$ The function that satisfies the equation. It is equal to $\lim_{n\to+\infty}(f^n(x))$ Then $^xa=Ψ^{-1}((Ψ(1)-τ)λ^x+τ)$

3. It is continuous among the reals (seems to) but can be discontinuous, especially in the complex plane.(I've found only one example)

4. Both properties are satisfied

5. Base : any numbers inside the Shell-Thorn region (basically all complex numbers whose infinite tetration converges. Such as $i$, or $\sqrt2$). And any complex hight which isn't an integer smaller than $-1$. (If the hight is a real greater than $e^{1/e}$, I can use a trick to make it work but it doesn't work with complex hights)

6. I don't know

The purpose of the post is to regroup every methods, and to explain them. So if there are any specialists in the field, it would be great! (Also because I don't understand the other methods than mine)