How to add timestamp to STDERR redirection

Question

In bash/ksh can we add timestamp to STDERR redirection?

E.g. myscript.sh 2> error.log

I want to get a timestamp written on the log too.

Answer 1

This thing: nohup myscript.sh 2> >( while read line; do echo "$(date): ${line}"; done > mystd.err ) < /dev/null &

Works as such but when I log out and log back in to the server, it does not work. that is mystd.err stop getting populated with stderr stream even though my process (myscript.sh here) still runs.

Does someone know how to get back the lost stderr in the mystd.err file back?