The symbol |z| means the modulus of z, and thus the length of the vector in the complex plane from the origin to z.
If z is purely real, then this vector lies along the real axis and so the length is the absolute value of z. As such, the absolute value function is the real-value special case of the modulus. Thinking of |z| as an absolute value when z is a non-real complex number is unhelpful because it is the absolute value that is the special case of a more general property, even though they are necessarily taught in the reverse order.
Since
where 
is a vector from 
to 
, it can be broken into perpendicular component vectors 
parallel to the real axis and 
parallel to the imaginary axis. Applying Pythagoras' Theorem shows that
} \\ &= |x|^2 + |i|^2|y|^2 \qquad \qquad \text{as $|z_1z_2| = |z_1||z_2|$} \\ &= |x|^2 + 1^2 \times |y|^2 \qquad \qquad \text{as $|i| = 1$} \\ &= x^2 + y^2 \qquad \qquad \text{as, for $k \in \mathbb{R}$, $|k|^2= k^2$, and as $x,\ y \in \mathbb{R}$ by definition} \end{align*})
Looking at the complex conjugates, we can see that
 \times \big(\,\overline{x + iy}\,\big) \\&= (x + iy)(x - iy) \\ &= x^2 + ixy - ixy - i^2y^2 \\ &= x^2 - (-1)y^2 \qquad \qquad \text{as $i^2 = -1$} \\ &= x^2 + y^2 \\ &= |z|^2 \end{align*})
If z is purely real, then this vector lies along the real axis and so the length is the absolute value of z. As such, the absolute value function is the real-value special case of the modulus. Thinking of |z| as an absolute value when z is a non-real complex number is unhelpful because it is the absolute value that is the special case of a more general property, even though they are necessarily taught in the reverse order.
Since
Looking at the complex conjugates, we can see that