Solution:

Adding $1 + i$ to both sides, the left side becomes $$z^4 + 4z^3 i - 6z^2 - 4zi + 1 = (z + i)^4,$$ so $(z + i)^4 = 1 + i.$

The four solutions for $w = z + i$ are equally spaced on a circle of radius $|1 + i|^{1/4} = (2^{1/2})^{1/4} = 2^{1/8},$ and they form a square. Subtracting $i$ merely translates it.

A square inscribed in a circle of radius $2^{1/8}$ has diagonal $2 \cdot 2^{1/8} = 2^{9/8},$ so its side is $\tfrac{2^{9/8}}{\sqrt{2}} = 2^{5/8}.$

The area is $$\left(2^{5/8}\right)^2 = 2^{5/4}.$$

Thus, D is the correct answer.