The typical way to represent a Bessel function of first kind is $ J_{\alpha}(z)$, i.e. $ J_{\alpha}: \mathbb{C}\to \mathbb{C}$.
Is there any good reason that prevents us to write it as a function of two variables $J(\alpha,z)$, i.e. $ J: \mathbb{C}^2\to \mathbb{C}$ ?
On
It depends very much on the context:
In physics, Bessel functions often appear in problems with cylindrical or spherical symmetry where $z$ plays the role of appropriate radial coordinate. The parameter $\alpha$ is usually fixed by some quantization condition but is not directly related to physical coordinates.
In the theory of linear ODEs with rational coefficients, the Bessel equation corresponds to a model rank 2 system with prescribed structure of singularities on the Riemann sphere $\mathbb C\mathbb P^1$ (one regular and one ramified irregular singularity of Poincaré rank 1). Here $z$ corresponds to a canonical coordinate on $\mathbb C\mathbb P^1$ whereas $\alpha$ encodes monodromy properties of the system. So again, the nature of the two parameters is rather different.
On the other hand, in addition to 2nd order differential equation w.r.t. $z$, Bessel function also satisfies 2nd order difference equation (recurrence relation) w.r.t. $\alpha$. It is possible to think of these equations in the same way. However, in order to see the analogy between $z$ and $\alpha$ more clearly, one needs to discretize the first parameter and go into the world of $q$-special functions.
No, there is nothing to prevent considering $J_\alpha(z)$ as a function of both $\alpha$ and $z$. However, not quite on $\mathbb C^2$: it has a branch point at $z=0$ if $\alpha$ is not an integer.