It's more to do with emphasis, not specifically anything to do with "with a new video". The same would be true of "Here I am" by itself. Because "am" needs a certain emphasis in this phrase, the contraction "I'm" is not used.
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AnonymousWhy is it called "Here I am with a new video" and not "Here I'm with a new video" ?Note also that we cannot contract 'be' with omitted participles and complements:
GPYA better example would be if you were showing photos to your friend and said "Here I'm with my mother" (= "Here" + "I'm with my mother"). .Please tell me if they can be understood this way as well.