Using fullscreen mode for splited window
tl;dr: Zoom the pane with <prefix>z or resize-pane -Z.
The default function for <prefix>w is choose-tree -w (version 2.6+; in older versions it was choose-window). These are displayed in the pane, so zoom the pane to fill the window temporarily.
Combining the two operations on a single keybind should be fairly straightforward. This can go in your .tmux.conf or into a running session (<prefix>:):
# replace default window-chooser: zoom first
bind-key -T prefix w resize-pane -Z \; choose-tree -w
See man tmux:
choose-tree [-GNsw] [-F format] [-f filter]
[-O sort-order] [-t target-pane] [template]
Put a pane into tree mode, where a session, window or pane may
be chosen interactively from a list. -s starts with sessions
collapsed and -w with windows collapsed. [...]