Applying a function a non-integer amount of times

The simplest case of fractional iteration is when $\,f(x) := c\cdot x\,$ with fixed point $0$, convergence is linear and the converging factor $\,0<c<1.\,$ Then the fractional iteration $\,f^{(t)}(x) = c^t\cdot x.\,$ Complications arise if $c$ does not satisfy the bounds. That is, the exponentiation $\,c^t\,$ is multivalend. The general case reduces to the simplest case using conjugation of composition with the convergence factor still the first derivative at the fixed point. Similarly for quadratic or higher orders of convergence.

In the example of $\,f(x):=\log(x),\,$ $\,f(w + x) \approx w + c\,x\,$ where $\,w\approx 0.318+1.337i\,$ and the convergence factor $\, c := f'(w)\,$ is $\approx 0.168-0.707i.\,$ Now we want an expansion for $g(x)$ so that $$\log(w+g(x)) = w+g(c\cdot x)$$ where the conjugating function $g(x)$ has a power series expansion $$g(x) \approx x + (-0.151 -0.296i)x^2 +(-0.036+0.098i)x^3 + (-.025-0.017i)x^4 + O(x^5).$$