As far as I know, the most visible way that complex numbers show up "in the real world" is as sine waves. Sine waves of a given frequency can be thought of as complex numbers. Adding together two sine waves corresponds to adding the corresponding complex numbers. Convoluting two sine waves corresponds to multiplying the corresponding complex numbers.

Since every analog signal can be thought of as a sum or integral of sine waves of different frequencies, an analog signal can be represented as a collection of complex numbers, one corresponding to the sinusoid at each frequency. This is what the Fourier transform is. Since convolution of analog signals corresponds to multiplication of their Fourier transforms, now a lot of the stuff we know about multiplication is applicable to convolution as well.