Let $I=\int_{-\infty}^{\infty} f(x) dx$ exist, state when $J=\int_{-\infty}^{\infty} f(x+ic) dx=I, c \in R $

Question

Currently, a question in MSE: How do I evaluate this type of integral that involve complex parameters? gives rise to the following query:

Let $I=\int_{-\infty}^{\infty} f(x) dx$ exist and let $J=\int_{-\infty}^{\infty} f(x+ic) dx, c \in R. $ The question is when $J=I.$

To re-emphasize that the argument of the integrand in $J$ has an imaginary shift to that of $I$. Can there be a simple and straight forward criterion on $f(x)$ and $c$, so that $J=I$?