TITLE: Algebraically closed field with characteristics $0$ versus $\mathbb{C}$
QUESTION [9 upvotes]: We know that $\mathbb C$ is an algebraically closed field with characteristic $0$.
It seems that if a proposition that can be expressed in the language of first-order logic is true for an algebraically closed field with characteristic $0$, then it is true for $\mathbb C$ (and for every algebraically closed field with characteristic $0$).
I am looking for interesting results that would be true over $\mathbb C$ but not true over some algebraically closed field with characteristic $0$.
I am more interested in "applied" results than results in field theory.
Additional comment: The initial question is vague (but on purpose). Here is what triggered that question in my mind. Sudbery proved the following result: Let $f(z)$ be a polynomial of degree $N$ with complex coefficients, and let $f^{(r)}(z)$ denote the $r$th derivative of $f(z)$. If $f(z)$ has two or more distinct roots, then $\prod_{i=0}^{N}f^{(r)}(z)$ has at least $N+1$ distinct roots. This result does not seem to generalize easily to algebraically closed fields with characteristic $0$.

REPLY [8 votes]: I (too) find the original question quite vague: there are so many differences between $\overline{\mathbb{Q}}$ and $\mathbb{C}$ that I wouldn't know where to begin.  What is more interesting is that by model theory the first order theories of these fields agree.  
Compare this with the "Lefschetz Principle" in algebraic geometry: if $K_1$ is any algebraically closed field of characteristic zero and infinite absolute transcendence degree, then "algebraic geometry is the same over $K_1$ as it is over $\mathbb{C}$".  The statement in quotation marks is not precise but should definitely be construed to mean much more than just equality of the first order theories, as classical algebraic geometers well know (this is the principle behind Weil's universal domains).  Model theorists have tried to enunciate the Lefschetz Principle more precisely, although to the best of my knowledge never in a fully definitive way.
With regard to your "Additional comment": for every positive integer $N$ you have written down a first order statement $P(N)$ in the language of fields which Sudbery showed holds true over $\mathbb{C}$ using, in part, convexity arguments.  I claim that it follows immediately from the completeness of $\operatorname{ACF}_0$ that for all $N \in \mathbb{Z}^+$, $P(N)$ holds over any algebraically closed field of characteristic zero (even though there is apparently no sensible notion of "convexity" in this context -- i.e., the model theory is actually helping out here!).  Further, by Ax's Transfer Principle, for each fixed $N$, there must exist a positive integer $M(N)$ such that for all primes $p \geq M(N)$, the statement $P(N)$ holds over any algebraically closed field of characteristic $p$.  You can find (what I hope is) a very detailed, elementary presentation of this transfer principle in $\S 3.5$ of these notes.
Now, let me draw your attention to the following MathSciNet Review:

MR1152642 (92j:11144)
Bartocci, Umberto; Vipera, Maria Cristina
On the Gauss-Lucas lemma in positive characteristic. (Italian summary)
Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 82 (1988), no. 2, 211–216 (1989).
If $f(x)$ is a nonconstant polynomial with coefficients in the field of complex numbers of degree $n$, then it has at least one root α; for all k in the range μ≤k≤n, where μ is the multiplicity of α, we have f(k)(α)/=0 (such a root is called a "free'' root of f). This is a consequence of the Gauss-Lucas lemma which was proved by A. Sudbery [Bull. London Math. Soc. 5 (1973), 13--17; MR0320288 (47 #8827)].
   Sudbery conjectured that this property remains true for polynomials with coefficients in a field of positive characteristic p>n. Bartocci [Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 56 (1974), no. 6, 856--858; MR0389865 (52 #10695)] proved that this conjecture is false, but true for p>C(n). In this paper it is shown that, if $n<p<2n−2$, then there exist polynomials which do not have free roots at all.  (Reviewed by M. Marden)

That is, the part of Sudbery's conjecture that follows immediately from standard model theory was proven (not using model theory, so far as I know) in a paper published one year later.  Much later it was shown that the stronger qualitative form of Sudbery's conjecture is very far from being true.  (This situation is somewhat similar to what happened with Artin's Conjecture on $p$-adic fields: see $\S 7$ of the same notes.)