If a black hole was next to another black hole and both where equal size what would happen? Would they swallow each other and make a bigger black hole?

I’ll echo the other replies that the gravitational waves from black hole mergers have been detected by LIGO. In fact, the 2017 Nobel Prize in physics was awarded to members of this collaboration specifically for this feat.

We haven’t (yet) seen a pair of black holes collide using light directly, but the gravitational waves have been perfectly consistent with general relativity calculations. Here’s a video from LIGO that shows what one of these simulations looks like, for a simulation that reproduces a detected gravitational wave.

As an aside, right around the time the LIGO team was awarded the Nobel prize, they detected the collision of a pair of neutron stars. They alerted the astronomy community to the direction they saw the signal from, and within a day there were telescope observations of light from the kilonova that resulted from the collision. Ultimately various sensors recorded optical light, infrared, ultraviolet, gamma rays, and radio waves being emitted from the explosion. The hope is that someday we’ll get lucky enough to see similar photon signatures from a black hole merger!

We know that black hole mergers are a thing, as LIGO has detected gravity waves from these exact events.

To get too much more specific, we need to ponder the mass of the black holes and their distance of separation.

You did specify that these black holes were of equal size. They would orbit each other, potentially for billions of years, just like any two other massive objects and how these orbits behaved would depend on their mass, orbital distance, relative velocity and the gravitational influence of any other large bodies. For example, two 30 solar mass black holes orbiting close to Sagittarius A* (our galaxy's central super massive black hole) would have a very different orbital pattern from the same two black holes orbiting each other in intergalactic space.

Black holes merge. It doesn't matter what size they are. It's not that the bigger one eats the smaller one they just merge.

Yes! And this happening is exactly what allowed gravity waves to first be detected

Yeah