Here is a proof that I think deserves to be recorded here somewhere. Of the proofs already listed it is closest to Pushkar's proof, Lucas Culler's proof, and Gian Maria Dall'Ara's highest-upvoted proof (arguably it is "just" a more complicated version of this third proof); I like the clear intuition it offers for why FTA is true, and it constructs a different interesting auxiliary space than the spaces constructed in the other proofs. Daniel Litt wrote a version of it down in 2011, and it is related to Arnold's proof of the unsolvability of the quintic which dates to 1963.
The idea is to argue that
the space of polynomials of degree $n$ with distinct roots (over a splitting field) is connected, and
the number of roots (over $\mathbb{C}$) of a polynomial with distinct roots is a locally constant function on this space.
Since e.g. $p(z) = z^n - 1$ has $n$ distinct roots over $\mathbb{C}$, it follows that every polynomial of degree $n$ with distinct roots has $n$ distinct roots over $\mathbb{C}$. This establishes the FTA in general, because if $f(z)$ is an arbitrary polynomial we can apply this result to its squarefree part $\frac{f(z)}{\gcd(f(z), f'(z))}$.
In more detail, the space $p(z) = z^n + a_{n-1} z^{n-1} + \dots a_0$ of monic polynomials of degree $n$ is $\mathbb{C}^n$. The discriminant $\Delta$ (the resultant $\text{res}(p, p')$) is a polynomial on $\mathbb{C}^n$ which is zero on a polynomial $p(z)$ iff it has repeated roots (over a splitting field). The complement $\mathbb{C}^n \setminus \{ \Delta = 0 \}$ is connected (this is the part of the argument that fails over $\mathbb{R}$), either abstractly because it has codimension $2$ or concretely by an argument involving complex lines.
Now it suffices to argue that the number of roots of a polynomial is a locally constant function on this complement. Here I am actually not sure what the cleanest way to finish the argument is. This can be done using the inverse function theorem; a calculation of a suitable Jacobian shows that in fact, on $\mathbb{C}^n \setminus \{ \Delta = 0 \}$ the roots of a polynomial are $C^{\infty}$ functions of its coefficients, so we can just follow the (distinct) roots around as we take a smooth path in $\mathbb{C}^n \setminus \{ \Delta = 0 \}$.
The argument above implies that $\mathbb{C}^n \setminus \{ \Delta = 0 \}$ can be identified with the unordered configuration space $\text{Conf}_n(\mathbb{C})$ of $n$ distinct points in $\mathbb{C}$. This is famously a classifying space for the braid group $B_n$. Essentially what we have to show to prove FTA is that the map sending a polynomial to its roots is a local system on this space. Its monodromy is given by the natural quotient map $B_n \to S_n$, and in particular is surjective; that is, moving around in a loop in $\mathbb{C}^n \setminus \{ \Delta = 0 \}$ can produce any permutation of a (separable) polynomial's roots. This is the starting point of Arnold's topological proof of the unsolvability of the quintic.
One can, as with several other proofs, see clearly where this argument fails over $\mathbb{R}$: the complement $\mathbb{R}^n \setminus \{ \Delta = 0 \}$ is no longer connected, because now $\{ \Delta = 0 \}$ only has codimension $1$ rather than codimension $2$. Its connected components are labeled by the number of real roots (equivalently, the number of complex roots); we can visualize moving around in this space by visualizing the roots, which are either real and so confined to the real line or which exist in complex conjugate pairs. At the discriminant locus $\Delta = 0$ two real roots can collide and split into a conjugate pair of complex roots, or two conjugate complex roots can collide on $\mathbb{R}$ and split into two real roots.
Gian Maria Dall'Ara's proof is a simplification of this proof where instead of varying all the coefficients we only vary the constant term. But if we do the proof this way we don't get to talk about braid group monodromy!