Self Portraits

Self portraits can be very intriguing in many ways, and can be much more than just a picture of someone. For instance, Frida Kahlo painted many very surreal and emotional self portraits with a lot of different layers:

 She uses a lot of very graphic detail and surreal imagery to get across something she may not have been able to with mere words. Her use of nature and blood is present in a lot of her work, but it is widely varied and all different.
Other artists don't have so much imagery in their self portraits, but they are more about experimenting with style and what they can do to make their portraits distinct.

These self portraits by Van Gogh and Picasso are perfect examples of this. While Van Gogh's isn't as stylized as Picasso's, the coloration and textures are not particularly realistic. And Picasso's os clearly very stylized.
To visually depict your sense of self, you don't even really need to draw yourself; the self portrait could be just a visual representation of different facets of your personality or what's important to you. They need not be your own face, it could be anything so long as it is some sort of representation of the artist.
If I were to make a self portrait, it would depend on my emotional state what it looked like, but generally I like to use very quick, jagged lines, scribbles and not much blank space. I try to fill up areas and make them a bit purposefully busy a lot  of the time. I also generally prefer monochrome, or black and white with one or two other bright, clashing colors used in small quantities. I like to represent the fact that my mind is kind of always in a lot of different places at once with the business of some of my art (also, maybe I'll draw a cat with me because I like them. There isn't much of a deeper meaning to that one, I just have a lot of cats).

Michael Landy

Michael Landy is a British visual and performance artist whose most famous pieces include destroying all of his belongings, something he described as being one of the best things he'd ever done, and Art Bin, an installation where he created an enormous bin that artists could throw their "failures" into. I find these pieces, as well as his non-performance art pieces to be rather intriguing, and well worth a look at whether or not you are a fan of performance art. His visual art can be very complex and aesthetically interesting, and have an all together whimsical feeling that make them distinct.

These pieces, for example, are very complicated and busy looking, but not in a negative way. They are a mess of different seemingly random things all stacked together and demand a lot of attention to fully take them in, but not in a tiring way. I'd say his art is worth taking a look at.