What happens to the event horizon of a black hole when another black hole comes close?

The event horizon is not a physical item but merely a place where the pull of gravity is exactly that of the speed of light.

If two black holes were near each other, the event horizon would bulge AWAY from their closest areas.

Other gravitational pulls will cancel out the pull of gravity. For instance: a person will weigh 0.0003384 percent less when our Moon is directly overhead as opposed to when it is on the horizon.

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The event horizon of the two black holes are summed in a complex way, the resulting event horizon is not equal to the sum of the event horizon of the two black holes! However the resulting event horizon would be that of a super-massive black hole the rigid body dynamics of this super-massive black hole would depend on the fact whether the original black holes were rotational or not!

Actually, the EH would bulge out towards the other black hole, which means that it would be more curved (and less flat).

The equations are in the last link, but unfortunately, I couldn't find any pictures or videos that match exactly.

You might like you at a number of graphic images or computer simulations of such a thing. Here is one that will lead to other links. It's something some take seriously and are working on...

http://www.youtube.com/watch?v=DxYwdgHpbKM