What is the bandwidth of an imaginary convolution?
However, we are in the imaginary frequency domain, so what would the
  bandwidth of this filter be?


I'm not sure why you're having difficulty with the fact that the frequency domain signal is, in this case, imaginary.  The symmetries of the Fourier transform are usually taught early on in signal processing courses:


If the time domain signal is real and odd, e.g., \$\sin\omega
   t\$, the associated frequency domain signal is imaginary and odd.
If the time domain signal is real and even, e.g., \$\cos\omega
   t\$, the associated frequency domain signal is real and even.


But, for bandwidth calculation, you're interested in the magnitude in the frequency domain (think Bode magnitude plot).  Since your frequency domain signal is pure imaginary, it couldn't be any easier; simply remove the j factor.

If, however, your frequency domain signal were complex, you would need to multiply the function by its conjugate and take the square root in order to find the magnitude.