The cost function of LS algorithm

I'm trying to understand the derivation of LS algorithm for channel estimation in OFDM.

Could anyone explain why do we need complex conj transpose in the below equation?

Because when a quantity can be complex, and even when it is just real, the absolute squared difference $$|f-g|^2$$ can be expressed in both domain (complex and real) as:
$$|f-g|^2 = (f-g)^H(f-g)$$ and of course this is correct as well for reals. This setting is often related to Hilbert spaces.