In this paper, we explicitly explore the endomorphism monoid of the circulant complete graph \(K(n, 4)\). We demonstrate that \(Aut(K(n,4)) \cong D_n\), the dihedral group of degree \(n\). Furthermore, we show that \(K(n,4)\cong D_n\) is unretractive for \(n = 4m , 4m +2\) (\(m \geq 2\)), and that \(End(K(n,4)) = qEnd(K(n,4))\) and \(sEnd(K(n,4)) = Aut(K(n,4))\) when \(n = 4m, 4m + 2\) (\(m \geq 2\)). Additionally, we prove that \(End(K(4m,4))\) is regular and \(End(K(4m + 2,4))\) is completely regular. We also solve some enumerative problems concerning \(End(K(n,4))\) are solved.