vim 8 and neovim have pretty functional in-built terminals (:term). You can try that if you're using one of these.
nmap gz :term zsh<CR>
:nnoremap <C-z> :term<Return>
Ctrl-Z only gets converted to SIGTSTP if the terminal has ISIG set. For example, try `stty -isig; sleep 5` and try to suspend or interrupt the sleep command.