complex numbers and fourier transform
there are issues.  given this convention for the continuous Fourier transform (and inverse)

$$ \mathscr{F} \Big\{ x(t) \Big\} \triangleq X(f) \triangleq \int\limits_{-\infty}^{+\infty} x(t) e^{-j 2 \pi f t} \ \mathrm{d}t $$

$$ \mathscr{F}^{-1} \Big\{ X(f) \Big\} \triangleq x(t) = \int\limits_{-\infty}^{+\infty} X(f) e^{+j 2 \pi f t} \ \mathrm{d}f $$

it changes the path of integration from the real axis to something else.  this comes up when using this fact:

$$ \mathscr{F} \Big\{ e^{- \pi t^2} \Big\} = e^{- \pi f^2} $$

to get, along with using scaling, this result:

$$ \mathscr{F} \Big\{ e^{j \pi t^2} \Big\} = \sqrt{j} \, e^{-j \pi f^2} $$

which is a linearly-swept "chirp" and it's spectrum.