I will apply the classical cox regression and time dependent cox regression, there are missing observations in my data. should i delete them or Should I use imputation methods?
Absent further information about the nature of the missingness, imputation will be the best approach. The advantages of imputation are explained in detail in Stef van Buuren's book. Even if the values are "missing completely at random" in the technical sense explained there, you are likely to lose power from omitting potentially useful cases. If there are any associations between the missingness and outcome, you run a risk of bias.
That said, sometimes the complete-cases approach can work adequately. You won't know, however, unless you try. In one early application of multiple imputation to survival data, the imputation didn't change the survival estimates much; the poor prediction of the missing covariate values pretty much outweighed the expected advantages. The trick will be developing a suitable imputation approach, which will require some careful application of your knowledge of the subject matter.