I don't believe that my thought patterns have gotten richer or more complex to any significant degree. While I overthink things, and overguess what people's intentions might be if I hear something that could possibly be taken wrong, this is more easily attributable to insecurity, low confidence, and/or a greater level of cynicism, and/or a better understanding of humanity as a whole (listed from most to least likely). The insecurity bit is an especially good note.
I will mention that I make a much better conversationalist than I used to, and that friends are thereby easier to make. I didn't have a friend to my name up until soon after I started HRT... Although I've been realizing that perhaps I talk too much. It doesn't bother me, but I worry about how others might react to what might be an overwhelming personality, at times.
My emotions are quite different. While I could be happy before, it was less common, and less potent, than the happiness that I see now. Sadness may have shifted, however this could just be because I've acquired the ability to cry. When I'm sad and angry about something, it seems to be felt to a much greater degree; in fact, it feels as if the emotions themselves will start to damage things in my environment. I could say that it is likely that my emotions have gotten stronger, and of course, that I'm happy more often.
Additionally, my emotions appear to be harder to mask. I used to be able to stop feeling whenever I wanted... especially around others. I don't find that I have as much of an ability to do this as I used to. This could be because of the seemingly greater intensity of my feelings.
My musical and other tastes may have changed, too. I know I don't listen to the music I used to, but it is difficult to say whether that is hormones, or just the natural progression of a human being. I tend to prefer less-hardcore music, and smoother vocals. Loving darker themes in music has not changed all that much, tough. As far as others go, I'm finding it really hard to stay interested in video games these days. It would be a shame if I lost that addiction, as I really enjoy(ed) them!
Overall, though, I don't feel that I'm becoming more complicated. If anything, I feel more simple. My emotions are less hidden, my thought patterns have remained the same, new insecurities aside... my tastes have amounted to a net loss in interest in things around me, and all those things have for redemption is a greater sense of happiness, and a new affinity for socializing.

Sorry for the essay

Edit: I forgot to add one troubling thing. I can't say whether it's hormones or growing up, but I think I feel a lot less lucid than I used to be. That is, like my actions and thoughts are a lot more automated than they used to be. I don't have any idea what this would be about... though I believe the downward decline in my lucidity pre-dates the beginning of HRT for me. Perhaps it's all the electronics, video games, and whatnot.