Summary: In this paper, we consider the existence of multiple solutions of the following critical nonlocal elliptic equations with magnetic field: \[ \begin{cases} (-i\nabla-A(x))^2u = \lambda |u|^{p-2}u+\bigg(\int_{\Omega}\frac{|u(y)|^{2^*_\alpha}}{|x-y|^{\alpha}}dy\bigg)|u|^{2^*_\alpha-2}u\quad\text{ in}\quad \Omega,\\ u = 0\quad \partial\Omega, \end{cases}\tag{1} \] where \(i\) is imaginary unit, \( N\geq4\), \(2^*_\alpha = \frac{2N-\alpha}{N-2}\) with \(0<\alpha<4\), \(\lambda>0\) and \(2\leq p<2^* = \frac{2N}{N-2} \). Suppose the magnetic vector potential \(A(x) = (A_1(x), A_2(x)\dots, A_N(x))\) is real and local Hölder continuous. We show by the Ljusternik-Schnirelman theory that (1) has at least \(cat_\Omega(\Omega)\) nontrivial solutions for \(\lambda\) small.