You could do that, it definately works. If following your way, we would use the difference of squares. But, that would be long and messy.
Instead, we can say that x^2=y, and swap y^2 for x^4.
We get y^2=-4.
So, y is -2i and 2i.
Now, we see that x^2 is -2i and 2i. After solving, the final answers are: x=1+i, x=-1-i, x=-1+i, x=1-i
:)