1. Interpret the geometry

The circle is $x^2+y^2=1,$ so its center is $O(0,0)$ and radius is $1$.

The chord is the line $y=mx+1 \,\Rightarrow\, mx-y+1=0.$

We are told that this chord subtends an angle $45^\circ$ in the major segment.

For a chord $AB$, if a point lies on the major segment, then the angle subtended by the chord at that point equals half the measure of the minor arc $AB$. Hence the corresponding central angle subtending the minor arc is: $\angle AOB = 2\times 45^\circ = 90^\circ.$

So the chord must subtend a central angle of $90^\circ$.

2. Use chord-length / distance-from-center relation

For a circle of radius $R=1$, if a chord subtends central angle $\theta$, then its distance from the center is $d=R\cos\frac{\theta}{2}.$

Here $\theta=90^\circ$, so $d=\cos 45^\circ=\frac{1}{\sqrt2}.$

Thus the distance from the origin to the line $mx-y+1=0$ must be $\frac{1}{\sqrt2}$.

3. Compute the distance from center to the chord

Distance from $(0,0)$ to the line $mx-y+1=0$ is $\frac{|1|}{\sqrt{m^2+(-1)^2}}=\frac{1}{\sqrt{m^2+1}}.$

Set this equal to $\frac{1}{\sqrt2}$: $\frac{1}{\sqrt{m^2+1}}=\frac{1}{\sqrt2}.$

Squaring both sides, $m^2+1=2$ $m^2=1$ $m=\pm 1.$

4. Match with options

The obtained values are $m=1 \quad \text{or} \quad m=-1.$

Check options:

• A: $2\pm\sqrt2$ gives values not equal to $\pm1$
• B: $-2\pm\sqrt2$ gives values not equal to $\pm1$
• C: $-1\pm\sqrt2$ gives values $\sqrt2-1$ and $-(1+\sqrt2)$, not equal to $\pm1$

So none of A, B, C matches.

Therefore the correct option is $\boxed{\text{D: none of these}}.$

5. Compare with stored answer

Stored correct answer is C, but our derivation gives D.

Hence, I disagree with the stored answer.

A likely reason: the standard inscribed-angle theorem gives central angle $90^\circ$, leading unambiguously to $m=\pm1$ and therefore option D.