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Is this the watch you're describing? From 1941, but I can't see any indication of a 'weird looking contraption at the base'.
"The Lindbergh Hour Angle Watch was designed
by Col. Charles A. Lindbergh as a navigational
aid for pilots. Used in conjunction with a
sextant, a nautical almanac and a radio set for
receiving the time signal, it made possible the
rapid determination of the Greenwich hour angle,
i.e., longitude"
The crown at 2 o'clock is for rotating the inner dial.
Click on image to enlarge.
photo credit (and quote) - www.farfo.com. Hope this doesn't break WTF rules for linking to non-sponsor, but I should credit the source.
"The Lindbergh Hour Angle Watch was designed
by Col. Charles A. Lindbergh as a navigational
aid for pilots. Used in conjunction with a
sextant, a nautical almanac and a radio set for
receiving the time signal, it made possible the
rapid determination of the Greenwich hour angle,
i.e., longitude"
The crown at 2 o'clock is for rotating the inner dial.
Click on image to enlarge.

photo credit (and quote) - www.farfo.com. Hope this doesn't break WTF rules for linking to non-sponsor, but I should credit the source.