"My other me."
"Ich" has two meanings, 1. I 2. self. The second meaning is Neutrum. Therefore, the adjective must be in Neutrum form, which in case of Diese, is Dieses. In short, for Maskulin Nominativ nouns, you use Dieser, for Maskulin Akkusativ nouns, Diesen, for Feminin use Diese and for Neutrum use Dieses.
"Ich" does not directly refer to "me" (although "Petar and me are going to the cinema" translates to "Petar und ich gehen ins Kino". in other cases, "me" translates to "mich" or "mir" depending on the context and the surrounding words in the sentence).
As I said, "I" and "self" are the two main meanings of "Ich", and the second one is always capitalized, since it's a noun, and nouns are always, always, capitalized in German. "Das Ich" means "self".
Of course, the first one is also capitalized when it's placed in the beginning of the sentence:
1- Ich gehe nach Bett.
2- Wer, ich??
That's my two cents :)
And then again, it doesn't necessarily mean that it's worthless, but it's just a "hint", telling us that there are, not very few, people out there that use this very sentence, which are neither language learners, nor stupid (take a look at http://www.amazon.com/Mein-Anderes-Ich-Gunter-Gabriel/dp/B004L21EKI for instance, which is a music album). Regarding the degree of usefulness to teach a language learner, you might be right. But the thing is that they probably want us to learn the other meaning of Ich (as a noun) in this question, and this sentence just does the job.