It is known that the pseudo-exponential map $K \to K^\times$ for $K$ a pseudo-exponential field (of cardinality $2^{\aleph_0}$) in the sense of Zilber is invariant under many field automorphisms, (probably there are $2^{2^{\aleph_0}}$ -such automorphisms which preserve the pseudo-exponential). That is, there are many $\sigma:K \to K$ ($\sigma$ is a field automorphism) such that $\sigma \circ \exp (x) = \exp \circ \sigma(x)$ for all $x \in K$, where $\exp$ is the pseudo-exponential.
It is not known whether $(K,\exp)$ is isomorphic to $(\mathbb{C},e^x)$ (the complex numbers field equipped with the standard exponentiation). But assume we could be able to prove that the complex exponential field has also automorphisms other than the identity and complex conjugation.
What would be the possible applications of such a result? Would it be interesting in its own right?
