Understanding the grammatical structure of 「まったくどうかしてるな 僕も」

Question

From the first episode of Death Note:

まったくどうかしてるな 僕も

apparently means

Seriously, what am I doing?

I was wondering if someone could help me parse this sentence out grammatically. Things that seem weird to me (as a beginner):

1. According to the dictionary, "まったく" is apparently an Adverb or a na-adjective. Since there's no な after it, I presume it's an adverb? Yet there's no verb in this sentence either?

2. The last word is "僕も", which is a noun, correct? Why is there no verb or copular at the end of this sentence?

3. "どうかしてる" is said to be an expression according to my dictionary. So what function does the な after it serve?

Answer 1

• 僕も is at the end of the sentence because it was added as an afterthought. Semantically the sentence is the same as 僕もまったくどうかしてるな.
• This も is an "exclamatory-も" described in this question: Odd use of も has me stumped 「ソウルにいるのも今夜きりです」
• どうかしている (literally something like "acting somehow") is a set phrase that euphemistically means "something is wrong (with someone)", "nuts", "insane", etc., depending on the context.
• This まったく is basically an adverbial intensifier meaning "totally", but it's also used like an interjection. "Seriously" or "Indeed" is a possible translation.
• な is a sentence-end particle for (self-)confirmation or adding emotion. See: When is "na" used at the end of a sentence? and Role of な in 嬉しいな

So the whole sentence means "Something's totally wrong with me(, huh?)".