Poisson noise on ##a_{\ell m}## complex number: real or complex?

[fab13]

TL;DR Summary

I try to get clarifications about the Poisson's noise with spherical harmonics of Legendre transformation

1) In a cosmology context, when I add a centered Poisson noise on ##a_{\ell m}## and I take the definition of a ##C_{\ell}## this way :

##C_{\ell}=\dfrac{1}{2\ell+1} \sum_{m=-\ell}^{+\ell} \left(a_{\ell m}+\bar{a}_{\ell m}^{p}\right)\left(a_{\ell m}+\bar{a}_{\ell m}^{p}\right)^* ##

Is Poisson noise a complex number or is it simply a real number ? knowing that variance of Poisson is equal in my case :

##\text{Var}(\bar{a}_{\ell m}^{p}) = \dfrac{1}{n_{gal}\,f_{sky}}## where ##n_{gal}## the density of galaxies and ##f_{sky}## the fraction of sky observed.

I work with fluctuations of matter density (not temperature fluctuations).

2) What is the variance of real part and imaginary part of an ##a_{\ell m}## : usually, one says that :

##\text{Var}(a_{\ell m}) = C_{\ell}## but given the fact that ##a_{\ell m}## is a complex number, we could say that :

##\text{Var}(\text{Re}(a_{\ell m}))## has a variance equal to ##\dfrac{C_\ell}{2}##

and

##\text{Var}(\text{Im}(a_{\ell m}))## has a variance equal to ##\dfrac{C_\ell}{2}##

since :

##\begin{aligned}
& \left|a_{\ell m}\right|^2=\operatorname{Re}\left(a_{\ell m}\right)^2+\operatorname{Im}\left(a_{\ell m}\right)^2 \\
& E\left[\left|a_{\ell m}\right|^2\right]=E\left[\operatorname{Re}\left(a_{\ell m}\right)^2\right]+E\left[\operatorname{Im}\left(a_{\ell m}\right)^2\right]=C_{\ell}
\end{aligned}##

Is it correct ?

Any clarification is welcome.