jmc

Let $z=a+bi.$ Then $$a+bi+\sqrt{a^2+b^2}=2+8i.$$Equating the imaginary parts, we get $b=8.$ Equating the real parts, we get $a+\sqrt{a^2+64}=2.$ Then $$\sqrt{a^2+64}=2-a.$$Squaring both sides, we get $a^2+64=(2-a{)}^2=a^2-4a+4.$ Solving, we find $a=-15.$ Therefore, $z=\boxed{-15+8i}.$