Disown won't take -h option

bash, zsh and ksh93 are the shells that have a disown command. Of the three, bash is the only one that supports a h option. Without -h it emultates zsh behavior (remove from the job table), while with -h it emulates ksh93 behavior (will not send it a SIGHUP upon exit, but doesn't remove it from the job table so you can still bg or fg or kill it).

You could emulate that behavior with zsh, by doing:

typeset -A nohup
trap 'disown %${(k)^nohup}' EXIT
trap 'for i (${(k)nohup}) (($+jobstate[i])) || unset "nohup[$i]"' CHLD

So the nohup associative array holds the list of jobs that are not to be sent a SIGHUP upon exit. So, instead of disown -h %1, you'd write nohup[1]=.

See also setopt nohup to not send SIGHUP to any job upon exit.

disown on its own is sufficient to let the program keep running after you disconnect:

$ command_running_forever
^Z
zsh: suspended  command_running_forever
$ bg
[1]  + continued  command_running_forever
$ jobs
[1]  + running    command_running_forever
$ disown %1
$ logout

disown -h uses a bash argument which causes the job to remain on the job table and does not receive a SIGHUP when bash receives a SIGHUP (according to bash's help disown). This argument is not available with zsh.