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We compute the contributions to rare top decays $t\rightarrow cZ$ and$t\rightarrow ch_1$ from the scalar sector in the 2HDM type III with CPviolation, where $h_1$ is the Standard Model Higgs boson. The branching ratiofor $BR(t\rightarrow c Z)$ and $BR(t\rightarrow c h_1)$ are obtained as afunction of the model parameters. In particular, the $BR(t\rightarrow c Z)$ canincrease its value up to $10^{-3}$ for $\tan\beta= 1$ and masses for theadditional Higgs bosons of $m_{h_2,h_3,H^\pm}\sim 0.5$ TeV. Meanwhile$BR(t\rightarrow c h_1)$ can reach values of the order of $\sim10^{-2}$. Weconstrain the model parameters (mixing angles of the neutral scalar fields inthe CP violation context and $\tan\beta$) using the reported values of thesignal strengths $R_{XX}$ and $b \rightarrow s \gamma$ process.

SM Higgs boson and $t\rightarrow cZ$ decays in the 2HDM type III with CP violation