What is this. After playing BtS about 15 minutes, a long brown pole comes out of each warrior's head and moves up and down like a long thin brown beam.

I'm on a Dell 410 XPS with Nvidia graphic card.

What is this and what can I do?

stwils

What model is your graphics card?
Getting the newest drivers for it might help if you haven't done so already. As a temporary fix you could try changing the graphics settings.

Edit: Is it happening only to warriors?

My graphics card is 256 Nvidia 7900 GS. Dell updated the driver about two weeks ago.

No, the pole is also coming out of a settler's head. I had only built a few units when this happened.

I set my graphics to only show one unit instead of the ring of three.

What should I do?

(Thank you for your response.)

stwils

Just offering suggestions to help, I myself have crashes so I know the feeling of civ gone wrong.

It sounds like a driver problem but if that didn't fix it I don't really know what to do. I would try to reinstall directx9 first and checking again about the latest graphics card driver. Also further lowering graphics might help. If all else fails there's the option of reinstalling civ.
The latest official nvidia updates are found here:
http://www.nvidia.com/Download/Find.aspx?lang=en-us

I've been searching around looking for solutions to your problem. A lot of renderer problems but can't find anything about spikes coming out of unit models so it must be a very uncommon problem.

Haven't seen any spikes problems in many years.

I think I will uninstall the game. Then reinstall but WITHOUT the patch. Do you think that might help?

I also played one game a few weeks ago and it seemed to thunder and lightening - screen flickered. So I quit that game.

Maybe uninstalling the game and reinstalling it without the patch would help.

What do you think?

stwils

Have you tried shutting your computer off and letting it cool down, then checked to see if the lines are there? If they disappear and come back after another 15 minutes or so, I would say your graphics card is failing.

Does it happen with anything else or JUST the settler.
Does it happen in any other games after a period of time?
Do you clean your computer of dust every so often?
Does your computer get excessively hot?

No. It doesn't happen in other games. And I had not played more that 20 minutes so I don't think the computer was hot.

Do you think the patch could have caused the problem?

stwils

It's possible the patch caused it. 3.13 introduced a lot of weird hardware problems for people. But that distortion you describe sounds like your card is overheated, especially that screen thundering/flickering. I don't think it's overheating from just civ, it's most likely something wrong on the card itself like screws or the heatsink, or even dust is covering it.
Usually the graphics malfunctions like yours are caused by an overheated card from the above mentioned problems, or people overclocking. But you mention it doesn't happen on other games...I think it's worth a try checking if the card has heat problems or needs a dusting. Trying without the patch is also a good idea.