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.