Let $$(z)$$ represent the principal argument of the complex number $$z$$. The, $$|z| = 3$$ and $$\arg(z-1) - \arg(z+1) = \frac{\pi}{4}$$ intersect:

Given,

$$|z|=3$$

which represents the circle

$$x^2+y^2=9$$

Also,

$$\arg(z-1)-\arg(z+1)=\frac{\pi}{4}$$

$$\Rightarrow \arg\left(\frac{z-1}{z+1}\right)=\frac{\pi}{4}$$

This represents the locus of points subtending an angle $$\frac{\pi}{4}$$ at the points

$$(-1,0)\ \text{and}\ (1,0)$$

Hence, the locus is a circle having chord length

$$PQ=2$$

Using

$$PQ=2R\sin\theta$$

we get

$$2=2R\sin\frac{\pi}{4}$$

$$R=\sqrt2$$

Let the center be $$(0,a)$$.

Then,

$$a^2+1=(\sqrt2)^2$$

$$a=1$$

So, the locus circle has center

$$(0,1)$$

and radius

$$\sqrt2$$

Now, distance between the centers of the two circles is

$$1$$

and

$$1+\sqrt2<3$$

Hence, the smaller circle lies completely inside the circle

$$|z|=3$$

Therefore, the two curves do not intersect.

Hence, no value of $$z$$ satisfies both equations.

Therefore, the correct answer is

$$\boxed{\text{Option C: Nowhere}}$$