Are the DFT/FFT multiplications in the fourier domain complex multiplications?
Yes, this is correct - the DFT/FFT of a real signal (for example an image) is not necessarily real; so if you want to compute a convolution through multiplications in the Fourier domain, the coefficients to multiply will be complex numbers; and will thus cost 4 real multiplications and 2 real additions if a naive implementation is used (unless the architecture you are writing code for has hardware support for complex multiplications).