I am working with multiple imputation and read almost universally how the number of imputations needed is relatively small. So it is recommended to use $M=5$ or thereabouts, that is the default is Stef Van Buuren's
mice package in
R. The rationale for this is not explained, but if we are to accept
mice as being efficient and unbiased, it seems reasonable that even with MCMC error, the resulting estimates could be considered reasonable.
However, I find that when I set different seeds, create imputed datasets, then calculate the results, the results will differ out at the second or third decimal place, and I am finding that primary inference may even differ.
For reporting (such as in academic manuscripts), should estimates be free of MCMC error, or is it okay as long as the results are replicable and the Seed is prespecified rather than chosen to provide "ideal" results.