Do complex iterates of functions have any meaning? Using a method explained in this answer to How to solve $f(f(x)) = \cos(x)$?, it is possible to calculate not only integer and real iterates of functions but also complex ones, for example, the $i$-th iterate, where $i=\sqrt{-1}$. Here are graphs of the $i$-th iterates of some common functions (the blue is the real part and the red curve is the imaginary part):
$$\arctan^{[i]}(x)$$

$$\sin^{[i]}(x)$$

So the question is whether there is any intuitive meaning to complex iterates, especially, say, $i$-th iterates of functions?
 A: I'm discussing this from the view of iterated exponentiation (although the technical process should be the same with other functions as well).
If you can use the Schroeder-function for the continuous iteration, then the iteration-height-parameter (say "h") goes into the exponent of some basis (the log of the fixpoint, often denoted as $ \small \lambda$ ). Imaginary heights h then switch the value of the Schroeder-function to the negative; this allows then to extend the iteration, in some sense, "beyond infinite height".

For instance, use base $ \small b = \sqrt 2 $ for iterated exponentiation, $ \small z_0=x, z_1=b^x , z_2=b^{b^x}, \ldots $. Then

*

*if you begin at, say, $ \small z_0=x=1$ you can iterate to infinite height to approach the limit at $ \small z_\infty = 2^{\small ^-}$ .

*if you start at $ \small z_0=x=3$ you can approach $ \small z_\infty = 2^{\small ^+}$ or even $ \small z_{-\infty}=4^{\small ^-}$ .

*if you start at $ \small z_0=x=5$ you can approach $ \small z_{-\infty} = 4^{\small ^+}$ or even $ \small z_{\infty}=  \infty$ .

$ \implies $ You cannot iterate from a value $ \small  z_m<2 $ to a value $ \small  2 < z_w < 4 $ using real heights, even when infinite.
But if you use the imaginary unit height you iterate directly from $ \small z_m=1$ to something like $  \small z_{m+i}=2.4 $:

*

*Assume again $ \small z_0=1$. Then the value of the Schröder-function (which is assumed to be normed to have the powerseries
$ \small \sigma(x)= 1x+\sum_{k>1} a_k x^k $ ) is about $ \small s=-0.316049330525 $.

*Then with height say $h=1$ gives $ \small \sigma°^{-1}( \lambda^1 s)\cdot 2 +2=b=\sqrt 2$ because that is the iteration of height 1 (in the exponent of $ \small \lambda$ ). (Remark: the "circle"-super-postfix $\sigma°^{-1}$ means       the functional inverse, not the reciprocal)

*If we replace that exponent by
$  \small h_w = i \cdot {\pi \over  \ln \lambda } $ then we get $  \small \sigma°^{-1}( \lambda^{h_w} s) \cdot 2 +2=2.46791405022...$ which is thus, in some sense, "beyond infinity" with respect to the iteration height.


late update I add a picture to illustrate the previous statements.
This is picture, where I studied the application of imaginary heights, using the base for exponentiation $b=\sqrt2$. It has the attracting real fixpoint $t=2$.
As an example, look at the left side, with $z_0=1 + 0\cdot î$. Using iteration with real heights (here in steps of $1/10$ ) we move rightwards to $z_1=b^{z_0}=b = 1.414...$ and by more iterations more towards the fixpoint $t =2+ 0 î$. This is indicated by the orange arrows.
Note that because $t$ is a fixpoint, we cannot arrive at points on the real axis more to the right hand!
But using imaginary heights, iterations move from $z_0 $ to $z_h$ on the indicated circular curve (computed data are in steps of $0.1 { \pi \over \ln \beta} î$ see legend), which is indicated by the blue arrow.
This iteration does not go towards the fixpoint, but repeats to cycle around it. On that cycling the trajectory crosses the real axis beyond the fixpoint.
(Legend: the circular curves which connect the computed iteration-values of imaginary heights are Excel-cubic-splines and thus only very rough approximations of the true continuous iterations)

A: Complex iterates of linear operators on Banach spaces, in particular imaginary iterates, have quite a lot of meaning in operator theory and they have applications to, among others, abstract parabolic equations.
Given a sectorial operator $A$, i.e. a linear closed injective densely defined operator $A$ on a Banach space $X$ such that $(-\infty,0)$ is contained in the resolvent set of $A$ and
$$\sup_{t<0}\|t(t-A)^{-1}\|$$ is finite, we say that $A$ admits bounded imaginary powers if the operators $(A^{is})_{s\in\mathbb{R}}$ form a $C_0$-group of bounded operators on $X$ where $A^{is}$ is defined via a suitable functional calculus.
As far as I know there is no reasonable partial differential operator on $L^p(\Omega)$ with $1<p<\infty$ known not to admit bounded imaginary powers (at least after a suitable translation along the real axis); the situation changes once we pass to $\Psi$DOs, though.
If an operator $A$ admits bounded imaginary powers this has remarkable consequences:


*

*If $X$ is a $UMD$-space and there is $\theta\in (0,\frac{\pi}{2})$ such that the group $(A^{is})_{s\in\mathbb{R}}$ satisfies $\|A^{is}\|\leq Ce^{\theta |s|}$ for all $s\in\mathbb{R}$ then the operator $A$ has the maximal regularity property by a result of Dore and Venni.

*The domain of the complex powers $A^z$ of $A$ for $\Re z\geq 0$ can be obtained using complex interpolation: $$D(A^z)=\left[X,D(A^k)\right]_{\frac{\Re z}{k}}$$ for $k\in\mathbb{N}$ with $k>\Re z$.

*If $X$ is a Hilbert space then the functional calculus $f\mapsto f(A)$ for bounded holomorphic $f$ is continuous with respect to the norm topology.


A good source for this and related aspects of operator theory is the book Functional calculus for sectorial operators by Markus Haase.
A: The difficult case is around a fixed point of a function with derivative one. Irvine Noel Baker, 1932-2001, studied these from the viewpoint of formal power series with complex coefficients, beginning with some $ f(z) = z + a_{m+1} z^{m+1} + \ldots, \; a_{m+1} \neq 0.$ He changed the question to finding those 
$$ f_\lambda(z) = z  + \lambda a_{m+1} z^{m+1} + \sum_{n = m+2}^\infty b_n(\lambda) z^n$$ which commute with $f.$ For a given $f = f_0,$ there may or may not be any other $f_\lambda$ such that the power series is convergent near $z=0.$ The big theorem, with one case taken care of by his student Liverpool, is that the set of $\lambda$ for which $f_\lambda(z)$ converges near $0$ is one of three sets: (a) $\{ 0 \},$  (b) with some fixed $N \in \mathbb Z,$ the fractions $\{m/N, \; \mbox{all} \; m \in \mathbb Z\},$  or $\mathbb C$ itself. In the final case, where any complex $\lambda$ is allowed, Baker called the function $f$ embeddable, saying that the function is embeddable in a continuous group of analytic iterates. 
In case (b) there is some minimal $1/N$th iterate which cannot be further, um, divided. So there may be half-iterates of something without there being any one-third iterates.
My summary would be that Baker makes it quite sensible to talk about an $i$ iterate. The conceptual switch from trying to do half iterates to asking what formal power series commute with a given formal power series makes the whole thing tractable.  
Oh, original articles and books posted at 
http://zakuski.utsa.edu/~jagy/other.html 
EDIT: I found some of my notes from 2010. From what I can make out, the only example that we expect to be really pleasant is the family of linear fractional transformations
$$      f_\lambda(z) = \frac{z}{1 + \lambda z}                  $$
which all comute with each other, and nothing worse happens than a pole for each one at $z = -1 / \lambda. $ Note the group law $f_\lambda \circ f_\gamma = f_{\lambda + \gamma}$
I felt that all other embeddable families were essentially that, just take some holomorphic $h(z)$ with $h(0) = 0$ and $h'(0) = 1$ and get the very similar
$$    f_\lambda(z) = h^{-1} \left(  \frac{h(z)}{1 + \lambda h(z)}   \right),       $$ with Fatou coordinate 
$$ \alpha(z) = \frac{1}{h(z)}. $$
There is a bootstrapping method for solving for the Fatou coordinate $\alpha(z)$ which is probably due to Ecalle. I also noted $ \beta(z) = \frac{- h^2(z)}{h'(z)}$ but I forget what $\beta$ was for. No, here we go, it is an explicit description in KCG on solving for the Fatou coordinate, pages 346-352,  Iterative functional equations by Marek Kuczma, Bogdan Choczewski and Roman Ger. In general $\beta(z) = 1 / \alpha'(z).$
Note, though, that we have now introduced possible bad behavior when either $h(z)$ or, more likely, $h^{-1}(z)$ are undefined, in short we have probably severely curtailed the region of $\mathbb C$ where things are working well.  
Edit toooo: the Fatou coordinate may be defined on only a sector out of the origin, anyway
$$ \alpha(f(z)) = \alpha(z) + 1.$$ Then we get a family (but maybe only in a sector) by
$$      f_\lambda(z) = \alpha^{-1}( \lambda + \alpha(z) ),          $$
where $f_1 = f$ in this recipe. So once again, as in the linear fractional transformations, we can plug in $\lambda = i.$
A: I'm kind of embarrassed to give such a simple answer, but since I don't think any of the answers are in this direction: there is a whole world of study in dynamical systems associated to acting groups beyond $\mathbb{Z}$. What I mean is that the "traditional" case of a single invertible map $T$ acting on a space $X$ can be thought of as an entire $(\mathbb{Z},+)$-action $\{T^n\}_{n \in \mathbb{Z}}$ (which just happens to be determined by the single map $T$).
If you view this way, then it's easy to picture generalizing to other groups; for any group $(G,\cdot)$, you can define a $(G,\cdot)$-dynamical system as coming from a $(G,\cdot)$-action $\{T^g\}_{g \in G}$ of self-maps of $X$. Then, if
$G = (\mathbb{C},+)$, it's easy to talk of complex iterates.
Your maps are not assumed invertible, but everything above works for semigroup actions too, so you could work with the upper-right quadant of $\mathbb{C}$.
Maybe what you want is more computational (and maybe other answers are doing this in a more direct way), but this is the first thing that came to mind as a dynamicist.
