I am not the only person who finds the 4 lines of

Quick search
[                  ] [go]
Enter search terms or a module,
class or function name. 

more eye-catching than the more useful
                                                      modules | index

in the corner. It is not even obvious what those are about. They are in black 'I am not a link' text until the mouse goes over them. So lets make them more prominent.

Second, no one finds the current, eye-catching Quick search very useful. For naive beginners, it seems to be off-putting.
The python-list thread is "python docs search for 'print'". So lets change it.

Combining these ideas, my revised suggestion is: Move the upper right entries to the sidebar above Quick search and expand and capitalize them so the sidebar would then look like

Module index
General index
Quick search
[             ]...

where the quick search is actually initially quick by first searching only the general index (Ezio's variation). (I am assuming that all modules are in the general index also). If something is found, it reports "Index entries found:" and only offers "Click here to do a full text search". If nothing is in the index, report "No index entries found, doing a full text search ..."

(This would still not be as good as the Windows doc, where the index tab starts with the entire index list and responds as each character is entered. That catches typos as mistyped. This would best be done by client-side javascript after loading the entire index list.)