Solve the equation $z^2 + leftvert z rightvert = 0$, where $z$ is a complex number.

I've tried solving this, but I'm stuck at one point.

Here's what I did:

Let $z = x + yi$, where $x, y \in \mathbf R$

Then , $(x + yi)^2 + \sqrt{x^2 + y^2} = 0$

Or, $x^2 + {(yi)}^2 + 2xyi + \sqrt{x^2 + y^2} = 0$

Or, $x^2 - y^2 + 2xyi + \sqrt{x^2 + y^2} = 0 + 0i$

Thus, $x^2 - y^2 + \sqrt{x^2 + y^2} = 0\qquad\qquad\qquad\qquad\qquad\qquad\ (i)$
and $2xy = 0 \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad(ii)$

If $2xy = 0$, then either $x = 0$ or $y = 0$

Now, if I take $x = 0$, and subsitute in $(i)$, I get either $y = 0$ or $y = 1$.

So far, so good, but if I take $y = 0$, and substitute in $(ii)$:

We have $x^2 + \sqrt{x^2} = 0$

so $x^2 = -\sqrt{x^2}$

or $x^2 = -x$

or $\frac{x^2}{x} = -1$

or $x = -1$

However, this solution doesn't satistfy the equation $x^2 + \sqrt{x^2}$ or the original equation.

What am I doing wrong here ?