  We present a methodology to recover cosmic microwave background (CMB)
polarization in which the quantity $P = Q+ iU$ is linearly combined at
different frequencies using complex coefficients. This is the most general
linear combination of the $Q$ and $U$ Stokes parameters which preserves the
physical coherence of the residual contribution on the CMB estimation. The
approach is applied to the internal linear combination (ILC) and the internal
template fitting (ITF) methodologies. The variance of $P$ of the resulting map
is minimized to compute the coefficients of the linear combination. One of the
key aspects of this procedure is that it serves to account for a global
frequency-dependent shift of the polarization phase. Although in the standard
case, in which no global E-B transference depending on frequency is expected in
the foreground components, minimizing $\left\langle |P|^2\right\rangle$ is
similar to minimizing $\left\langle Q^2\right\rangle$ and $\left\langle
U^2\right\rangle$ separately (as previous methodologies proceed), multiplying
$Q$ and $U$ by different coefficients induces arbitrary changes in the
polarization angle and it does not preserve the coherence between the spinorial
components. The approach is tested on simulations, obtaining a similar residual
level with respect to the one obtained with other implementations of the ILC,
and perceiving the polarization rotation of a toy model with the frequency
dependence of the Faraday rotation.
