Calculating poles with complex numbers and quadratic equation
You seem to be computing, on the square root side, something like, $a$ and $b$ being positive, $\sqrt{a-b}=\sqrt{a}-j\sqrt{b}$, which is wrong. The square root is not linear.

Here, the discriminant $\Delta = (0.8\sqrt{2})^2-4\times 0.8^2 = -2\times 0.8^2$ is negative. Hence, the roots are:

$$ z_r= \frac{-0.8\sqrt{2}\pm j\sqrt{-\Delta}}{2} = 0.4\sqrt{2}(-1 \pm j)$$

agreeing with Wolfram Alpha.