Notation confusion -- What is the correct operator for computation of the log-likelihood expression for complex valued data?
Without seeing your overall signal model we cannot really help. I suppose $y$ is your estimate? And hence, $L=L(y)$. Where is $h$ in your Likelihood equation? However, some general observations:


As Batman pointed out: likelihood corresponds to probability, which is necessarily a real value. So, if your equations yield complex values, you did something wrong. (Note that, depending on numerical accuracies etc. it can happen, that there is a very small imaginary part ($\approx 10^{-16}$) in your numeric computations. This part can be ignored.)
Commonly (but not always!) when in the real-domain you have a transpose in your equation, it changes to conjugate transpose (i.e. Hermitian operation) in the complex domain.
Noise variances commonly remain real-valued. However, noise realizations will be complex-valued. 
You should understand the expression in your exponential as the squared distance between the measurement and the estimate. In real domain, the squared distance of $x,y$ is equal to $(x-y)^2$. However, in the complex domain, it becomes $\|x-y\|^2=(x-y)(x-y)^*$.