  This paper deals with the following degenerate fractional Kirchhoff-type
system with magnetic fields and critical growth: $$ \left\{ \begin{array}{lll}
  -\mathfrak{M}(\|u\|_{s,A}^2)[(-\Delta)^s_Au+u] = G_u(|x|,|u|^2,|v|^2) +
\left(\mathcal{I}_\mu*|u|^{p^*}\right)|u|^{p^*-2}u \
&\mbox{in}\,\,\mathbb{R}^N,\\ \mathfrak{M}(\|v\|_{s,A})[(-\Delta)^s_Av+v] =
G_v(|x|,|u|^2,|v|^2) + \left(\mathcal{I}_\mu*|v|^{p^*}\right)|v|^{p^*-2}v \
&\mbox{in}\,\,\mathbb{R}^N, \end{array}\right. $$ where
$$\|u\|_{s,A}=\left(\iint_{\mathbb{R}^{2N}}\frac{|u(x)-e^{i(x-y)\cdot
A(\frac{x+y}{2})}u(y)|^2}{|x-y|^{N+2s}}dx
dy+\int_{\mathbb{R}^N}|u|^2dx\right)^{1/2},$$ and $(-\Delta)_{A}^s$ and $A$ are
called magnetic operator and magnetic potential, respectively.
$\mathfrak{M}:\mathbb{R}^{+}_{0}\rightarrow \mathbb{R}^{+}_0$ is a continuous
Kirchhoff function, $\mathcal{I}_\mu(x) = |x|^{N-\mu}$ with $0<\mu<N$,
$C^1$-function $G$ satisfies some suitable conditions, and $p^*
=\frac{N+\mu}{N-2s}$. We prove the multiplicity results for this problem using
the limit index theory. The novelty of our work is the appearance of
convolution terms and critical nonlinearities. To overcome the difficulty
caused by degenerate Kirchhoff function and critical nonlinearity, we introduce
several analytical tools and the fractional version concentration-compactness
principles which are useful tools for proving the compactness condition.
