Bash – Does zsh use canonical-mode for the terminal

• local is a builtin, not a keyword, so local files=(…) isn't parsed as an array assignment but as a string assignment. Write the assignment separately from the declaration. (Already found by llua, but note that you need to initialize files to the empty array or declare the variable with typeset -a, otherwise the array starts with a spurious empty element.)
• Zsh arrays are numbered from 1, not from 0 like in bash and ksh, so ${files[0]} must be written $files[1]. Alternatively, tell zsh to behave in a way that's more compatible with ksh and bash: put emulate -L ksh at the beginning of the function.
• Unless you go the emulate route, your _notes function will print zsh: no matches found: foo* if there is no completion for foo, because by default non-matching globs trigger an error. Add the glob qualifier N to get an empty array if there is no match, and test whether the array is empty.
• There is another error in your _notes function which affects notes in subdirectories: you must strip away the prefix up to the completion, so that if e.g. ~/notes/foo/bar exists and you type n b<TAB>, COMPREPLY is set to contain b, not foo/b.

If you want to keep a file that's readable by both bash and zsh:

type emulate >/dev/null 2>/dev/null || alias emulate=true
function n() {
  emulate -L ksh
  local arg; typeset -a files
  for arg; do files+=( ~/".notes/$arg" ); done
  ${EDITOR:-vi} "${files[@]}" 
}

function nls() {
  tree -CR --noreport $HOME/.notes | awk '{ 
      if (NF==1) print $1; 
      else if (NF==2) print $2; 
      else if (NF==3) printf "  %s\n", $3 
    }'
}

# TAB completion for notes
function _notes() {
  emulate -L ksh
  local x files
  files=($HOME/.notes/**/"$2"*)
  [[ -e ${files[0]} ]] || return 1
  COMPREPLY=()
  for x in "${files[@]}"; do
    COMPREPLY+=("$2${x#$HOME/.notes*/$2}")
  done
}
complete -o default -F _notes n

If you want to port your code to zsh:

function n() {
  local files
  files=(${@/#/~/.notes/})
  ${EDITOR:-vi} $files
}

function nls() {
  tree -CR --noreport $HOME/.notes | awk '{ 
      if (NF==1) print $1; 
      else if (NF==2) print $2; 
      else if (NF==3) printf "  %s\n", $3 
    }'
}

# TAB completion for notes
function _notes() {
  setopt local_options bare_glob_qual
  local files
  files=(~/.notes/**/$2*(N))
  ((#files)) && COMPREPLY=($2${^files##~/.notes*/$2})
}
complete -o default -F _notes n

Bash needs to put the terminal into character-at-a-time mode while it's waiting for you to type in a command line, so that you can edit the command line using emacs or vi-like editing characters. That's the mode you saw when you looked at the terminal's attributes from another terminal in your example.

Just before it runs a program (in your example, stty), bash puts the terminal back into canonical mode, where you have just a few special editing characters available courtesy of the operating system, such as backspace and Control-W, and basically the program gets input only after you type Enter.

When bash regains control, say after the program finishes or is suspended, it will put the terminal into character-at-a-time mode again.