 Solution

Put $z=x+iy$. Comparing imaginary parts gives $-y=1$, so $y=-1$. Comparing real parts then gives $\sqrt{x^2+1}=x$. The left side is positive, so squaring is legitimate only with $x\ge 0$, but even then it would give $x^2+1=x^2$. This contradiction proves there is no solution. Equivalently, with nonzero imaginary part the complex modulus is strictly greater than the real part.