= Solution
Let $T=\operatorname{Th}(\mathbb C)$ in the finite language of rings: it is the <set> of all <first-order sentences> true in the <field> of <complex numbers>. It is satisfiable, with <first-order model> $\mathbb C$, so the countable-language <Löwenheim-Skolem theorem> gives an at most countable <first-order model> $K$ of $T$.
All <field axioms> belong to $T$, hence $K$ is a <field>. For every positive <integer> $n$, the <first-order sentence> saying that there are at least $n$ distinct elements is also true in $\mathbb C$ and belongs to $T$. Thus $K$ cannot be finite, and is countably infinite.
Finally, for each <first-order sentence> $\sigma$, either $\sigma\in T$ or $\neg\sigma\in T$. Since $K$ satisfies all of $T$, it satisfies $\sigma$ exactly when $\mathbb C$ does. Therefore
$$
\boxed{|K|=\aleph_0,\qquad K\equiv\mathbb C.}
$$
<Elementary equivalence> concerns first-order <first-order sentences> and does not imply an <isomorphism>; the two <fields> have different <cardinalities>. The stronger downward form may even choose $K$ as a countable elementary subfield of $\mathbb C$.
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