Terminology question on band-shifting I am trying to figure out if there is anything more to the phrase 'complex band shifting' than just, a simple band shift. Disclaimer, this is NOT a communications problem, and I only say this so that some answers do get carried away with bit rates, etc.

Basically, let us say that I am interested in a signal, that exists in the passband, from frequencies $f_1$ through $f_2$. Let us further assume that I band pass filter this signal with a brick wall BPF. Then, I multiply it with a sine wave of frequency $f_1$. This operation shifts my signal 'down to baseband', and at base band, it now has a (single-sided) bandwidth of $f_2$ - $f_1$, (as it did before). (I can now LPF to get rid of the double frequency component). That is, my signal now exists from $0$ to $f_2$-$f_1$. 

To me, this is all there is to band shifting. That is, the key component is a multiplication with a real sinusoid. 

Why then, and what does, the phrase 'complex band shift' entail exactly? 

I understand that a multiplication with a complex sinusoid is done in comm systems because we then try to decipher phase from the I and Q, so I get that, but if I simply want to shift bands of a signal, (even to an arbitrary frequency), what is 'complex band shifting' and is that always necessary? Cant I just use a real sinusoid for general band shifting?

Thanks.
Complex band shifting is a little more flexible than real band shifting and filtering, and may be a little more efficient too, though if you change the sample rates when moving a signal to baseband that may not be true.

You are correct that you can move your signal to 0 to $f_2-f_1$ Hz in the method that you describe.  You could do the same thing in one step by multiplying the pass band signal by a complex sinusoid with a frequency of $-f_1 $.

Also, when you band shift with a real sinusoid you have to watch out for aliasing.  Band shifting with a complex sinusoid never causes aliasing (unless you reduce the sample rate at the same time, of course).