What is the mean power of a complex random variable?
The correct expression for the mean power of a complex random variable $\alpha=x+jy$ is
$$
\begin{align}
\bar P &amp;= \operatorname E\left[\alpha \alpha^*\right]\\
&amp;= \operatorname E\left[x^2 + y^2\right]\\
&amp;= \operatorname E\left[x^2\right] + \operatorname E\left[y^2\right]\\
&amp;= \bar P_\mathrm{x} + \bar P_\mathrm{y}
\end{align}
$$

In other words, the mean power of a complex random variable is the sum of the mean powers of its real and imaginary part, respectively. In contrast, expression no. 1 from your question evaluates to
$$
\begin{align}
\operatorname E\left[\alpha^2\right] &amp;= \operatorname E\left[x^2 + j2xy - y^2\right]\\
&amp;= \operatorname E\left[x^2\right] - \operatorname E\left[y^2\right] + \operatorname E\left[j2xy\right]\\
&amp;= \bar P_\mathrm{x} - \bar P_\mathrm{y} + j2\operatorname E\left[xy\right]
\end{align}
$$
Note that for the special case of $x$ and $y$ being uncorrelated $\operatorname E\left[xy\right]=E\left[x\right]E\left[y\right]$. If, in addition, $x$ or $y$ has zero mean $E\left[xy\right]=0$.