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).