It is essential to directly investigate the self couplings of gauge bosons in the Standard Model (SM) due to its non-Abelian nature, as these couplings play a significant role in comprehending the gauge structure of the model. The discrepancies between the Standard Model's expectations and the measured value of gauge boson self-couplings serve as strong evidence for new physics phenomena extending beyond the Standard Model. Such deviations provide valuable insights into the nature of new physics and potentially lead to a deeper understanding of fundamental particles and their interactions. This study examines the sensitivities of anomalous couplings associated with dimension-8 operators that affect the gamma gamma gamma gamma and Z gamma gamma gamma quartic vertices. The study focuses on the process e(-)gamma-* e(-)gamma gamma with the incoming photon under Weizs & auml;cker-Williams approximation at the stage-3 scenario of Compact Linear Collider (CLIC) that refers to a CoM energy of 3 TeV. Due to the CLIC options, we take into account both unpolarized and -/+ 80% polarized electron beam with the related integrated luminosities of L = 5, 4, 1 ab-1 under the systematic uncertainties of delta(sys) = 0, 3, 5. Obtained sensitivities on the anomalous quartic gauge couplings (aQGCs) for the process e(-)gamma-* e(-)gamma gamma at root s = 3 TeV and various polarizations are improved by a factor of 2- 200 times better for the couplings f(T,j)/A(4) compared with the experimental results reported by CMS Collaboration in Tumasyan et al. (JHEP 10:174, 2021).