$\chi$ is a real-valued function. This is part of the definition of the gauge transformation, since $U(1)$ is a one dimensional group. In general, when talking about gauge transformations in particle physics, group parameters are restricted to be real.

$\chi$ is a real-valued function. This is part of the definition of the gauge transformation, since $U(1)$ is a one (real) dimensional group. In general, when talking about gauge transformations in particle physics, group parameters are restricted to be real by convention.

In principle, I suppose you could perform a transformation on the wavefunction that looks just like a $U(1)$ gauge transformation except that the parameter can be complex. But the resulting group of transformations would not be $U(1)$, it would be some two-dimensional group, because a complex number parametrizes two dimensions.