What is the fourth contact for on the Bose A20 headset's mic plug?
This kind of plug has contacts, not pins, that one on the end of the mic plug's the PTT (Push To Talk) contact for a military spec microphone plug.

This helpful page explains it:

The first aircraft radios used carbon-granule microphones almost
identical to microphones used on telephones. However, a second
function was needed on the microphone in the form of a push-button to
key the transmitter and effect a changeover from receive to talk . . .
hence the name push-to-talk or PTT switch. Telephone companies not
only had pretty rugged microphones, they could offer a nifty 3-circuit
plug to go with it. To this day, the terms &quot;tip&quot;, &quot;ring&quot; and &quot;sleeve&quot;
coined by the telephone industry are used to identify the three
circuits. The addition of a &quot;ring&quot; connection allowed us to craft a
PTT microphone companion to the already common headset. Being smaller
than a headset plug made it easy to avoid plugging the headset into
the wrong jack and vise-versa.
This particular plug is a mil-spec critter and appears to have yet
another &quot;ring&quot; between the tip and ring connections I've noted . . .
this may or may not be visible on your particular microphone plug . .
. in this particular plug, it's simply a spacer to fill out the shape
an maintain commonality with the dimensions of other style plugs.

When you watch some WWII movies you see pilots touching their mics or boxes close to their neck to talk, that's because the PTT was wired in-line with the headset rather than being on the control column or throttle. The 'tip' was there to enable that, when the PTT was depressed an electrical connection was made between the tip and the sleeve, signalling the radio to switch to talk mode. Note that the movies aren't always right, some airplanes had a PTT button off the headset, but that's just one of many common inaccuracies.
Nowadays the PTT button is often wired on the other side of the plug so that the PTT button goes to the stick or yoke, but the principle is the same, it's just that the connection is made farther up the line.
The plug on your headset is a slightly older design with a spacer between the tip and the ring to ensure the two stay separate electrically. It isn't an actual contact because it doesn't go anywhere.