Is $|x| \cdot |x| = |x^2| = x^2$? 
Is $|x| \cdot |x| = |x^2| = x^2$ ?

I'm very sorry if this question is a duplicate but I couldn't find anything about it (most likely because it's wrong..). But I'm not sure if this is correct so I need to ask you.
$$|x| \cdot |x| = |x^2| \text{ should be alright}$$
Now my confusion starts. $x^2$ should be positive / neutral for any value. That would mean we can ignore the absolute value sign? On the other hand we could have that $|-x^2|$. But that would be a different thing than $|x^2|$, they are not equal to each other...? Please help me if I do this little thing wrong the entire task will be wrong. I got some thinking error here..
When there is the same question (I couldn't find one), please link me to it and I will delete this one immediately.

For real numbers, this is true. We have:
$$|x| = sgn(x) \cdot x$$
And thus:
$$|x| \cdot |x| = sgn(x)^2 \cdot x^2 = x^2$$
Also:
$$|x^2| = sgn(x^2) \cdot x^2 = x^2$$
