What is needed to base physics on iterated functions? Iterated systems are considered candidates for the foundation of physics. Iterated functions have an even stronger claim:
Arnold and Avez (1968)

Let $M$ be a smooth manifold, $\mu$ a measure on $M$ defined by a continuous positive density, $f^t:M \to M$ a one-parameter group of measure preserving diffeomorphisms. The collection $(M,\mu,f^t)$ is called a classical dynamical system.

I've been researching fractional iteration $f^n(z)$ where $f:\mathbb{C}\to\mathbb{C}$ and $n\in\mathbb{N}$ is extended to $n\in\mathbb{C}$ as a toy universe.
Question: what needs to be added to have a viable model of physics? I believe the generality plus the need for the chain rule makes Fréchet space a desirable extension. Boundaries need to be supported. Is there any way to generalize fixed points to invariant manifolds? What about physical boundaries like air foils? What else is needed or useful?
Derivation of a flow or fractionally iterated function.
Derivative of iterated function: $H(n,t)=D^nf^t(L)$
Fixed point: $H(0,t)=L$
Lyapunov multiplier: $H(1,t)=f'(L)^t$
$$H(n,t)=\sum_{r=0}^\infty(\sum_{k=1}^n \frac{f^{(k)}(L)}{k!} B_{n,k}(H(1,t-1),\ldots, H(n-k+1,t-1)))^r$$
$$f^t(x)=\sum_{k=0}^\infty\frac{1}{k!} H(k,t) (x - L)^k$$
While initially $t \in \mathbb{N}$, once a symmetry is added the result is consistent with $t \in \mathbb{C}$. Note: this is not a model of iterated functions, it is iterated functions in a power series form.

Mathematica code

Flow[f_, t_, x_, L_, order_ : 3] := Module[{},
H[0] = L;
H[1] = f'[L]^t ;
Do[
H[max] =
First[r[t] /.
RSolve[{r[0] == 0,
r[t] == Sum[
Derivative[k][f][L] BellY[max, k,
Table[H[j] /. t -> t - 1, {j, max}]], {k, 2, max}] +
f'[L] r[t - 1]}, r[t], t]],
{max, 2, order}];
Sum[1/k! H[k] (x - L)^k, {k, 0, order}]
];

 A: In a sense, physics is already modeled in this way.
A flow on phase space $M$ is just a one-parameter group, or group action, of the real numbers (time) upon the phase space. That is to say, the flow is a family of maps $\Phi^t : M \rightarrow M$, one labeled by each real $t$, that satisfy
$$\Phi^0 = \mathrm{id}_M$$
$$\Phi^t \circ \Phi^s = \Phi^{s + t}$$
which really is exactly the same as for an iterated function family of functions $f^n$ with $n \in \mathbb{N}$:
$$f^0 = \mathrm{id}_D$$
$$f^n \circ f^m = f^{m + n}$$
where $f: D \rightarrow D$ for some general domain $D$. And all models of physics known so far use flows on some suitable phase or configuration space, even quantum theory where that a Hilbert space, or else space of density operators on that space, are used (amongst other possible formalisms), and the flow advance map is the iterated unitary map.
The difference between what you are suggesting and how physics works now is that an iterated function family $\{ f^n \}_{n \in \mathbb{N}}$ does not generally extend uniquely to a continuous flow $\{ f^t \}_{t \in \mathbb{R}}$, if it even extends at all. Typically, either an extremely infinite number of possible extensions are available, since one can essentially "fill the gap" between some $n$ and its successor $n + 1$ in any way one wants (though once you have one $n$, the group action property then fixes it for the rest), or else something like constraints on the existence of limit cycles will obstruct extension.
Or to say it some way, "iterated functions" generalize most naturally to one-parameter groups. There is, in a sense, "more information" in a flow than in a discretely-iterated function, and thus you can go from the former to the latter, but not so the other way around. That is, every flow contains an iterated function within it, i.e. when you take only natural values of the index, but an iterated function does not in general uniquely determine a flow.
A: 
what needs to be added to have a viable model of physics?

How about a connection to observed reality? All you say is that you're studying iterated complex exponentials. That's just math.
What observable properties do you want to model? What quantities in your formalism will correspond to observable properties?
