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When we draw samples with complex dtype, we multiply their magnitude with sqrt(0.5). This is done because then a sample from a variance of 1 also has standard deviation of 1 on average. However, this is inconsistent with its Hamiltonian. If we instead regard a complex number as a pair of real numbers, then in order to reproduce the sample variance, we would need the vector (real, imag) to have covariance

 / 0.5  0   \
|           |
 \ 0    0.5 /   

However this covariance would lead to the Hamiltonian (real**2 + imag**2)*2 which is double of what we would get when using the complex likelihood with covariance 1.