  This paper is concerned with the multiplicity and concentration behavior of
nontrivial solutions for the following fractional Kirchhoff equation in
presence of a magnetic field: \begin{equation*}
\left(a\varepsilon^{2s}+b\varepsilon^{4s-3}
[u]_{A/\varepsilon}^{2}\right)(-\Delta)_{A/\varepsilon}^{s}u+V(x)u=f(|u|^{2})u
\quad \mbox{ in } \mathbb{R}^{3}, \end{equation*} where $\varepsilon>0$ is a
small parameter, $a, b>0$ are constants, $s\in (\frac{3}{4}, 1)$,
$(-\Delta)^{s}_{A}$ is the fractional magnetic Laplacian,
$A:\mathbb{R}^{3}\rightarrow \mathbb{R}^{3}$ is a smooth magnetic potential,
$V:\mathbb{R}^{3}\rightarrow \mathbb{R}$ is a positive continuous potential
having a local minimum and $f:\mathbb{R}\rightarrow \mathbb{R}$ is a $C^{1}$
subcritical nonlinearity. Applying penalization techniques and
Ljusternik-Schnirelman theory, we relate the number of nontrivial solutions
with the topology of the set where the potential $V$ attains its minimum.
