= Solution
The <Cantor-Schröder-Bernstein theorem> says that <injections> $X\to Y$ and $Y\to X$ imply a <bijection> $X\to Y$. Thus mutual cardinal inequalities imply equal cardinalities, without a <choice> assumption.
Write $\mathfrak c=2^{\aleph_0}=|\mathbb R|$. Standard <cardinal arithmetic> gives $\mathfrak c\cdot\mathfrak c=\mathfrak c$ and $\mathfrak c^{\aleph_0}=\mathfrak c$. The correspondence $x+iy\leftrightarrow(x,y)$ is a <bijection> $\mathbb C\leftrightarrow\mathbb R^2$. Hence
$$
\boxed{|\mathbb C|=\mathfrak c.}
$$
Equivalently, the real axis supplies the lower inequality and a coding of two real coordinates supplies the upper one; Cantor-Schröder-Bernstein then gives the equality.
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