I use bash as my shell and have long used:

.bashrc:
  promptFunc() {
    echo "$(date +%Y-%m-%d--%H-%M-%S) $(hostname) $PWD $(history 1)" \
      >> ~/.full_history
  }
  PROMPT_COMMAND=promptFunc

I was recommending this to folks at work, but they use zsh. In zsh you can get a lot of the way here by setting INC_APPEND_HISTORY (append to history immediately instead of waiting for the shell to exit), SAVEHIST=1000000000 (effectively don't limit the history size on disk), and EXTENDED_HISTORY (to store timestamps with history entries). But you risk losing most of your history if you ever accidentally invoke zsh without these set, and it doesn't write the directory you ran the command in (which is metadata I reference a lot).

Mike tweaked my snippet to run in zsh:

.zshrc:
  precmd() {
    echo "$(date +%Y-%m-%d--%H-%M-%S) $(hostname) $PWD $(history -1)" \
      >> ~/.full_history
  }

The two changes are that zsh uses precmd instead of PROMPT_COMMAND, and that you need history -1 instead of history 1.

You can still use the same histgrep command on both:

function histgrep {
  local n_lines=10
  if [[ "$1" =~ ^[0-9]*$ ]]; then
    n_lines="$1"
    shift
  fi
  grep "$@" ~/.full_history | tail -n "$n_lines"
}

Note that this doesn't replace your shell's built-in history tooling, and I wouldn't recommend turning that off. This just a very cheap additional layer of logging with additional metadata and less risk of accidental deletion.