As has been pointed out in the other answers, there are four complex numbers $z$ with the property that $z^4=i$. If you would like to understand the behavior exhibited by WolframAlpha, however, you should consider the fact that its square root function is (in fact) a function built on top of Mathematica's Sqrt
command. It returns exactly one number, namely
$$\sqrt{r e^{i\theta}} = \sqrt{r} e^{i\theta/2},$$
when $r\geq0$ and $-\pi<\theta\leq\pi.$
As a result, $\sqrt{\sqrt{i}}$ is uniquely determined to be
$$\sqrt{\sqrt{i}} = \cos \left(\frac{\pi }{8}\right)+i \sin \left(\frac{\pi }{8}\right) \approx 0.92388 + 0.382683 i.$$
That is exactly what Wolfram Alpha has told you in the first couple of pods.
When you say that Alpha says there are two roots, I assume that you are referring to the last couple of pods that refer to "all 2nd roots of $(-1)^{1/4}$". First off, those pods would be considered secondary. They arise because WolframAlpha typically produces secondary pods containing related information in an effort to respond to the various things that a human user might have meant in their input.
The reason that there are two is because sqrt(sqrt(i))
essentially parses as
sqrt(a complex number)
so you get the two roots of that complex number in the secondary pod. You would get the same result, if you type sqrt((1+i)/sqrt(2))
because
$$\sqrt{i} = (1+i)/\sqrt{2}.$$
Finally, if you want all four fourth roots of $i$, simply type
all fourth roots of i
You can get more information on how WolframAlpha deals with roots in this blog post.