Based on Johnny's obsessions with time and space and his comments to Bruno about once having almost seen to "the other side of the door," it seems that what Johnny is pursuing is a unified understanding of life. Johnny says he doesn't believe in God, so no religious concepts or doctrines can give him this understanding. Johnny does take drugs, which either derange his perceptions and enhance his belief in his vision. Johnny pursues this understanding by musing over the nature of a loaf of bread after it's cut: Is it the same in nature as before it was cut? He pursues understanding of time and space by contemplating the seemingly opposing natures of transportation on an elevator and subway.
Based on Johnny's music, which is the side of Johnny that Bruno restricts his biographical work to, Johnny is pursuing other things as well. He is pursuing the furthest limits of what music can do: what is on "the other side of the door" for his music to achieve. While in this pursuit, Johnny also pursues people, the people who are his friends and admirers, in order to expose them in their weaknesses, like when in conversation with Bruno he exposes doctors and scientists for their weakness, which is not knowing how the world works.