Can the Fourier transform be defined as an integration over $\mathbb C$ instead of $\mathbb R$? 
Can the Fourier transform of a whole function $f:\mathbb R\mapsto\mathbb C$ be defined as integration over $\mathbb C$ instead of $\mathbb R$ as well, such that 
  $$\tilde f(k) = \frac{\mathcal N}{2\pi} \int_{\mathbb C}f(x) e^{-ikx}\,dx,$$
  and (if so) what is the correct value of the normalization $\mathcal N$ for consistency with the $\mathbb R$-integration?

Given a whole function $f:\mathbb R\mapsto\mathbb C$, its Fourier transform
$$ \tilde f(k) = \frac1{2\pi}\int_{-\infty}^\infty f(x)e^{-ikx}\,dx$$
can be determined by other integration paths as well by using Cauchy's Residue Theorem, for example by shifting $x$ by an imaginary constant $ic$. Assuming a sufficiently fast decaying function for $\Re(x)\to\pm\infty$ (and using the fact that a whole function has no residues), this results in simply adding that constant to the integration boundaries, i.e.
$$ \tilde f(k) = \frac1{2\pi}\int_{-\infty+ic}^{\infty+ic} f(x)e^{-ikx}\,dx,$$
which can be expressed by substitution as well:
$$ \tilde f(k) = \frac1{2\pi}\int_{-\infty}^{\infty} f(x-ic)e^{-ik(x-ic)}\,dx.$$
One can then average over different values of $c$ to obtain
$$ \tilde f(k) = \frac1{2\pi\cdot 2T}\int_{-T}^{T}\int_{-\infty}^{\infty} f(x-ic)e^{-ik(x-ic)}\,dx\,dc.$$
Now my question boils down to
1) Since infinity is involved, is this equivalent to 
$$ \tilde f(k) = \frac1{4\pi T}\int_{\mathbb R\times[-iT,iT]} f(x_1+x_2)e^{-ik(x_1+x_2)}\,d^2x,$$
2) Can the limit $T\to\infty$ be taken such that $\mathbb R\times[-iT,iT]\to\mathbb C$
and 3) What would the correct normalization be?
 A: No, the Fourier transform of an entire function $f:\mathbb R\mapsto\mathbb C$ cannot be defined as integration over $\mathbb C$.
The comments already contain many hints why such a definition would be problematic. However, even the definition by integration over $\mathbb R$ is problematic for an entire function, as can be seen by the following part of the Paley–Wiener theorem:

An entire function $F$ on $\mathbb C^n$ is the Fourier–Laplace transform of distribution $v$ of compact support if and only if for all $z \in \mathbb C^n$, $$ |F(z)| \leq C (1 + |z|)^N e^{R| \mathfrak{Im} z|} $$
  for some constants $C$, $N$, $R$.

We learn from this that the Fourier transform of a tempered entire function is in general only a distribution, which implies that the Fourier integral will not be well defined for a general entire function.
If we look at ("tempered") functions holomorphic in an "appropriate" half-plane of $\mathbb C$ instead of entire functions, the (inverse) Laplace transform becomes the "appropriate" modification of the Fourier transform.
