Complex samples are inconsistent
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.