What is it called when you can see the sun setting and the full moon rising at the time?

As far as I know, there IS no special name for that.

It's fairly common--it's all straight lines, after all, so in order for the moon to be full the earth cannot be casting a shadow on it at all. That means that the sun and moon need to be on the same side of the earth at the same time.

As far as I know, it's called the end of a clear day.

Sorry, but that's about it.

As far as I know, it's called the end of a clear day.

Sorry, but that's about it.

How about "pretty sunset?"

It's quite common. More so in the fall (meaning around the autumnal equinox) than any other season because that is when at sunset the percentage of the ecliptic that is viewable above the horizon all at once is at a maximum.

If the Suns sets and the Moon is rising, it is a full Moon, if the Moon and Sun rise or set together, it is a new(no)Moon. Usually they are not in line with the Earth, when they are, it is an Eclipse. Either Lunar or Solar.

I'm afraid I must amend my earlier answer to this question. Upon further investigation I have learned that the moon does not travel precisely on the ecliptic, but rather its path is inclined to the ecliptic by about 5 degrees. This means that for northern hemisphere viewers, you are most likely to see the full moon rising as the sun is setting when the moon is at its maximum separation north of the ecliptic. Further, it helps to have the ecliptic tilted as far south as possible at the same time - and this does indeed occur in the fall as I noted earlier. So, while earlier I said that seeing the rising full moon with the setting sun is "common," I am now less certain of just how common it really is.

On a side note - we have a lunar (or solar) eclipses when the full (or new) moon occurs at the same time as the moon is close to the ecliptic. The points where the moon's orbit crosses the ecliptic are called the lunar nodes, and because of the precesssion of the moon's orbit the position of the nodes in the sky precesses along the ecliptic, taking about 18.6 years to make one full trip around the ecliptic. So the trick to predicting eclipses is to figure out when a full (or new) moon occurs within 1.5 degrees of the ascending or descending lunar node.

An eclipse?
I think that's what your asking anyway!

Not an eclipse unless we all line up, Earth Moon and Sun. (Not necessarily in that order).
I see the Sun Set and The Moon Rise Together about every 28 Days.
Alaska and Above latitudes and the Extreme lower latitudes that have 24 Hours of Sun, Or 24 Hours of Darkness, this does not apply.