If i have equation:
\begin{align} P = \left|\psi\right|^2 \end{align}
where $P$ is a probability and we know there is no negative probability. This means $P$ must belong to $\mathbb{R}$. If i want to calculate $|\psi|$ i can do it simply by sq. rt. the equation:
\begin{align} \left|\psi\right| = \sqrt{P} \end{align}
Is $\left|\psi\right|$ a real number? What about $\psi$ by itself? Please explain.
The absolute value of a complex number (sometimes called its modulus) is real by construction. If $\psi=\alpha+i\beta$ with $\alpha$ and $\beta$ real, then $|\psi|=\sqrt{\alpha^2+\beta^2}$.