For the record, let me write down the <A HREF="https://en.wikipedia.org/wiki/Mehler_kernel">Mehler kernel</A> for the square root of the Fourier transform,
$$K(x,y)=(2\pi)^{-1/2}\sqrt{1-i}\exp\left[{\tfrac{1}{2} i \left(x^2+y^2-2 \sqrt{2} x y\right)}\right].$$
As a check, I calculate
$$\int_{-\infty}^\infty K(x,z)K(z,y)\,dz=(2\pi)^{-1/2}e^{-ixy},$$
so indeed, application of the transform $K$ twice is equivalent to the Fourier transform.