Time for a trip to the 'Stealer' . . .

 

Take a pic of it..

 

No video found. The way you described it makes me think its something aftermarket. Also that it continues making noises with the battery disconnected makes me think its some kind of aftermarket alarm with its own power supply. A picture of it would certainly help.
The only round part i can think of is the SKIM antenna, which should sit around the ignition key cylinder, but it doesn´t look like a bulb.
Be aware that a weak battery can cause strange noises from relay´s and strange behaviour of the instruments, so check that out as well. Does the starter motor crank the engine with force, or does it crank slowly or not at all?

 

Does it have a blue blinking bulb? Do u have the Omega Alarm? Mine is clicking(like the blinker is on, but it's not) inside the Omega Control Box/Relay/Module..THINGY..I have some right side blinker, tail light issues too..They don't even flash when I Lock/set alarm..the other 3 do blink though..I already replaced the Flasher Multi Function Relay, so I'm thinking the alarm relay may be a contributor to my probs..Any Input from any of you guys MUCH Appreciated..Thanks, Charles
99LAREDO

 

Check your Battery. I recently had a similar problem and it turned out to be the battery. I replaced it with the biggest battery I could find. Interstate Mega Tron Plus. 1000 CA, 850 CCA.

 

A weak battery can make for some real interesting and strange operation of gauges, lights and clickity-clacking of relays... even after turning the ignition switch off.

 

Yeah Man, I just got a brand new 850CCA..no help w/ this issue though..Just missing the left turn signals, hazards(all of them) and this ticking from the Omega Alarm Module(sounds just like a blinker)..but it's not..STRANGE..ugh..