It's excellent that you've stumbled upon this now! It's something you may see in the future:
Technically, the root of a number isn't well defined. That is, the root "function" gives multiple values as its output. For the sake of simplicity, we usually restrict the output. We call all the roots of $1$ the roots of unity, for example. (It may be worth noting that, in mathematics, whenever you see the term unity, it usually refers to something that's equivalent to the number $1$).
I've stolen this definition directly from Wikipedia:
In mathematics, a root of unity [...] is any complex number that gives 1 when raised to some integer power, $n$.
There's something called Euler's Formula:
$$e^{i \theta} = \cos{\theta} + i\sin{\theta} \implies re^{i\theta} = r(\cos{\theta} + i \sin{\theta})$$ Which we use extensively in complex analysis.
There is another thing to know: We can write any complex number as $~re^{i \theta}~$, and thus write any complex number as $~r(\cos{\theta} + i\sin{\theta})~~^{[1]}$, by Euler's Formula above. Let's say $z$ is some arbitrary complex number. We can thus write it as $re^{i\theta}$. Now, take the $4^{th}$ root of this number:
$$\sqrt[4]{z} ~=~ z^{\frac{1}{4}} ~=~ (re^{i\theta})^{\frac{1}{4}} ~=~ re^{i \frac{\theta}{4}} ~=~ r\left(\cos{\frac{\theta}{4}} + i\sin{\frac{\theta}{4}} \right)$$
Remember from trigonometry: $\cos{\theta} = \cos{(\theta + 2\pi n)}$ where $n = 0, 1, 2, 3, ...$ This works the same way for $\sin{\theta}$. That is, you can add any multiple of $2\pi$ to $\sin$ or $\cos$ and the values don't actually change.
That means that the $4^{th}$ root of $z$ is $~r\left(\cos{\frac{\theta}{4}} + i\sin{\frac{\theta}{4}} \right)~$ and $~r\left(\cos{(\frac{\theta}{4} + 2\pi)} + i\sin{(\frac{\theta}{4} + 2\pi)} \right)~$ and $~r\left(\cos{(\frac{\theta}{4} + 4\pi)} + i\sin{(\frac{\theta}{4}} + 4\pi)\right)~$, etc.
It's true that you can add multiples of $2\pi$ to infinity, so there should be infinite roots of unity. The reason we only get four, in this case, is because—after adding enough multiples of $2\pi$—the values for $\sqrt[4]{z}$ start repeating themselves.
This is why treating the $4^{th}$ root of $i$ like a regular root seems to come out strangely. You were following all the mathematics that you know, but your proof actually stumbled onto something pretty in-depth!
$^{[1]}~~$This isn't rigorous, but a quick explanation: You can think of the different parts of a complex number, $~z = x + iy~$, as coordinates. It acts just like the regular $(x,y)$ system we usually use. Here, the $x$ of the complex number is the x-component and the $y$ is the y-component. So, $~4 + i7~$ is $(4,7)$ on the complex coordinate system. After doing this, we can perform geometric operations on our complex number much like we do on a regular coordinate system. Eventually we can derive the Euler's Formula stated above.
By the way, when the coordinates of a complex number $re^{i\theta}$ are mapped out onto the complex plane, the "repeated roots" that one gets all lie on the circumference of a circle whose radius is $r$. For the $4^{th}$ root, one gets four points. Continuing to add multiples of $2\pi$ will just repeat these four points over and over.
