It's easy to be jaded about FTA after mathematical history has run for another two centuries. Sure, it is not the number one most important result any more, or the center of any research program (though understanding the algebraic closure of Q could be considered as half of number theory). But consider the situation around 1800. In addition to solution of algebraic equations one had new methods of constructing numbers, using power series, integrals and other limits. Algebra and number theory dealt with the first situation, to a limited extent, and analysis showed that the second type of construction could be iterated but still stay within the same realm of numbers. There was still the possibility that solving equations with $\pi$ and $e$ as coefficients could require an entirely new type of super-transcendental analysis. Fundamental Theorem of Algebra is self-defeating in this sense: it shows that nothing more was needed than complex numbers. But this is not clear in a world where you don't know that FTA is true.
To get an idea what algebraic geometry looked like without complex numbers, look up Newton's classification of degree 3 algebraic curves in the plane, $P_3(x,y)=0$, using real coordinates. The reason this work is obscure today is that there are many dozens of cases compared to the complex projective version. As in Lie groups and topology, looking from the universal covering (C) downward, modulo some Galois-theoretic details (R), is usually easier than working from the bottom up.
Suppose you want to evaluate the integral, from $- \infty$ to $+\infty$, of a rational function (one with integer coefficients would illustrate the point). The answer will involve $\pi$ and the usual method for finding it will use the specific location of the roots in the complex plane, so is more specific to complex numbers than the existence of roots in an algebraic closure. There are some non-usual methods that stay entirely within the real numbers, but they are nonstandard because they are more complicated, and harder to understand and adapt to other problems.
Algebraic geometry in general has a transcendental part -- periods, Hodge theory, uniformization, etc -- that, in the present state of knowledge, cannot be fully substituted by algebraic methods over fields of characteristic 0 (or p). Sometimes Lefschetz principle or reduction to positive characteristic can be used, sometimes not, or the theory is unknown.