In the Windows help version of the docs that come with the Windows .msi
installer, the index pane to the left scrolls separately from the
content pane to the right.  Very nice for jumping around even to other docs.

In the html versions at docs.python.org, there is one scroll bar and the
index disappears when one moves very far down the page.  If sensibly
possible, decoupling index and context would be nice.  Please pardon my
ignorance if not.

Terry, I think you mean the Sidebar content, right? 
Yes, I agree with you. It would be desirable to have the Sidebar
Fixed, while we scroll the document (Like this:
http://www.w3.org/Style/CSS/) 

This has to be worked out in the Sphnix CSS.

I've experimented with a style variant that keeps the sidebar fixed on
the left side, however I did not manage to get it to show a separate
scrollbar.  Maybe I was just stupid though.

Something like this maybe?

div.sphinxsidebar {
    float: left;
    width: 230px;
    height: 100%;
    font-size: 90%
    /* add these: */
    margin-top: 30px;
    height: 100%;
    overflow: auto;
    position: fixed;
}

...

div.related {
    background-color: #133f52;
    color: #fff;
    width: 100%;
    line-height: 30px;
    font-size: 90%;
    /* add this: */
    position: fixed;
}

I tested it only with Firefox3 and the Webdeveloper plugin and it looks
ok. "position: fixed" doesn't work on IE6 and there could be other
problems, but it may be a good starting point.

See also #3143.
I'll try to do a proper patch to fix this issue.

"overflow: auto" should fix #4711 too (adding an horizontal scroll bar
at the bottom of the sidebar).

The attached patch makes the sidebar fixed and adds a scrollbar to scroll it separately when it's longer than the page.

To test the patch without rebuilding all the Docs you can do something like:
hg import --no-c <url-of-the-patch>
cp Doc/tools/sphinxext/static/basic.css Doc/build/html/_static/basic.css
firefox Doc/build/html/library/multiprocessing.html

Since the sidebar doesn't start at the top of the page because there's the header first, when the main page is scrolled down, some empty space is left over the sidebar.  I think this can't be fixed with css alone (unless some major changes are done to the layout), it could be fixed with js or the header could use position:fixed too.

In the root of the clone, with the doc already built in Doc/build (no need to rebuild it), do:
hg import --no-c http://bugs.python.org/file27887/issue4965.diff
cp Doc/tools/sphinxext/static/basic.css Doc/build/html/_static/basic.css
firefox file://`pwd`/Doc/build/html/library/multiprocessing.html