An alternate, purely dimensional way to see this: for a homogeneous half-space the impedance is \( Z = \sqrt{i\omega\mu_0\rho} \) where \(\rho\) is resistivity. Writing the complex number \(i\) in polar form, \( i = 1\angle 90^\circ \). Taking the square root of a complex number halves its argument (and square-roots its magnitude), so
\[ \sqrt{i} = 1\angle\left(\frac{90^\circ}{2}\right) = 1\angle 45^\circ \]
Since \(\omega\mu_0\rho\) is a purely real, positive scalar, it contributes zero phase — all of the phase of \(Z=\sqrt{i\omega\mu_0\rho}\) comes from the \(\sqrt{i}\) factor, i.e. exactly \(45^\circ\). This is why the MT phase curve for a uniform ground is a flat, frequency-independent horizontal line sitting exactly at \(45^\circ\) — any tilt of that line away from \(45^\circ\) in field data is telling you the subsurface is layered/heterogeneous, not uniform.
\(\boxed{45^\circ}\)