It is the idea that you have a multilayered and multifaceted personality that makes you unique and important. However this is what makes the issue of depth so complicated. Rather than simply accepting all of our own underlying qualities, we have this idea that we need to be validated by others which causes people to go out and try to seek validation and create depth in a sense so others will see them as a depthful and meaningful person. Unfortunately it is this quest to obtain depth that makes them so shallow in reality. This shallow attitude can even carry over into the world of dreams.
Dreaming of depth in general means you are thinking about your depth as a person. Sometimes people are described as either being deep or shallow. A deep person has a lot to offer to the world, a lot of important opinions and useful perspectives on the world while a shallow person has a lot to say that nobody actually wants to hear. Dreaming that you are deep or that you have depth is an indication that you could stand to be surrounded by a bit more depth in your life; it would be a great asset to you once you start out your new life.
To dream that you have reached astounding depths means that you are getting closer to figuring out what makes you tick. This is something that everyone searches for whether they want to admit it or not. Everyone wants to be a person that matters or a person that has some depth to them and nobody really knows who they are as well as they think they do. You need you need to accept how hard it is to know this before you can bear fruit on your quest.