Meaning of AWGN complex and real noise under BPSK Decision Scheme
It doesn't matter if the channel is complex or real. It's actually always complex, but since you are using BPSK, only the real part matters. How to add it is a mathematical trick. Say you need to generate an SNR of x in linear scale, and your signals have unit energy, then the noise variance must be 1/x. In other words, you don't generate real and complex parts for the noise, because then the SNR will be $1/(2x)$, which is not correct. You generate the real and complex parts only if the signal is complex, at which case the total noise variance gives the desired SNR. Hope this helps.