Interpret the given angle correctly

The ray makes an angle of $45^\circ$ with the interface.

Therefore, the angle of incidence with the normal is $i = 90^\circ - 45^\circ = 45^\circ.$

Use the given deviation

For refraction at a plane surface, if light goes from air to a denser medium, it bends towards the normal.

The deviation from the original direction is $\delta = i - r,$ where $r$ is the angle of refraction.

Given $\delta = 15^\circ,$ so $45^\circ - r = 15^\circ.$

Hence, $r = 30^\circ.$

Apply Snell's law

For air to medium, $n = \frac{\sin i}{\sin r} = \frac{\sin 45^\circ}{\sin 30^\circ}.$

Substitute values: $n = \frac{\tfrac{1}{\sqrt{2}}}{\tfrac{1}{2}} = \frac{2}{\sqrt{2}} = \sqrt{2} \approx 1.414.$

Match with the options

$1.414$ corresponds to Option C.

Comparison with stored answer

Stored correct answer: C

Derived answer: C

So they agree.