You are looking for a function $\displaystyle f(z) = \sum_{k \ge 0} \frac{c(k)}{k!} z^k$ satisfying
$$f(1) = 1$$
$$f(-1) = -1$$
$$f(i) = -i.$$
Infinitely many functions have this property. There is a unique quadratic polynomial $p(z)$ with this property (for example by Lagrange interpolation), and for any entire function $q(z)$ the function $p(z) + (x - 1)(x + 1)(x - i) q(z)$ has this property. In fact these are all entire functions with this property.
More generally I think the theory of interpolation by entire functions is fairly well-understood, but I don't know of a good reference. If the set of $z$ at which you fix the value of $f$ has a limit point, then $f$ is unique by the identity theorem. If the set of $z$ at which you fix the value of $f$ is countable and does not have a limit point, then $f$ is non-unique by Weierstrass factorization.