In the complex plane, the function
$$ z \to z^n$$
has a simple geometric interpretation. A complex number is specified by two pieces of data. A distance $r$ from the origin, and angle $\theta$ relative to the horizontal.
The function $z \to z^n$ sends a pair $(r,\theta)$ to the pair $(r^n, n\theta)$. In plain English, it raises the distance to the power $n$ and multiplies the angle by $n$. What is important to note here is that the angle takes values between $0$ and $2\pi$ and then loops back to $0$. To see why this gives rise to multiple roots, consider the equation
$$z^2 = 1$$
In our geometric understanding, this is asking: what complex numbers $(r,\theta)$ exist such that $r^2 = 1$ and $2\theta = 0$? Since $r$ is assumed positive, we must have $r=1$. But what about the angle? Clearly we can let $\theta = 0$, which corresponds to the solution $z=1$. But we can also let $\theta = \pi$, because if we double the angle $\pi$ we return to the angle $2\pi \equiv 0$. This corresponds to the solution $z=-1$. So the multiplicity of roots to the equation $z^2 = 1$ emerges from the fact that the angle resets every $2\pi$, and that $z \to z^2$ is composed of a stretching and a rotation. The same reasoning explains why $z^n = 1$ has multiple roots.
More generally, a polynomial like $z^8 - 3z^2 + 4$ can be thought of as describing some complicated geometric operation composed of stretching, translation and rotating. It is not then unreasonable to propose that there are multiple ways to arrive at the same destination via this operation.