Ham --- one that has to be relinquished, removed, taken out, killed, reject.
Aham -- one that cannot be relinquished, killed, rejected, removed, taken out etc.
I cannot free myself from me.
I can reject you, you can reject me, but no one can reject or be without Him, the God. So He is also aham. aham actually then means God only. Aspect outside which there is no independent identity or existence (astitva) of anything is called Aham. Aham is that which is the basis of everything. He is the Universal / infinite aham and we are all fragmental / subjective / minute / infinitesimal ahams.
Bhagawanta is in pindanda (jiva) and brahmanda (Universal womb). He is "aha" in both. The Bhagawantha in pindanda is referred to as aham, and in brahmanda is referred to as ahah (ahah samvatsaro vahnih analo dharini dharah). He is ahah, the one who gives life and light from within (the Jiva of the pindanda and the Surya of the Brahmanda).
Even Bible seems to say this, Exodus 3:14:
Then Moses said to God, “If I come to the people of Israel and say to them, ‘The God of your fathers has sent me to you,’ and they ask me, ‘What is his name?’ what shall I say to them?” God said to Moses, “I am who I am.” (((or, "I am that I am"). And he said, “Say this to the people of Israel, ‘I am has sent me to you.’”
Vedic Shatapatha BrahmaNa says: Yo-ham-asmi-so-smi "I am what I am".
Ormazd Yasht of the Zoroastrian religion declares: "My first name is ahmi, my last name is ahmi" (Essential Unity of All Religions). Ahmi infact means asmi.
So, "I am" or aham in this context does not mean "me" either in Bible or Upanishads. It infact refers to the Lord.
“parokSha priyA iva hi devAH pratyakSha dviShaH” (in bhR^ihadAraNyaka upaniShad 4.2.2), for the Gods love, as it were, what is mysterious, and hate what is manifest. So, Aham need not be direct. A true sadhaka needs to dig into details.
Asmi -- As + Mi -- Mi is Manam, jnyeyam. One that is to be known. As -- the all determining, the all inspiring one.
So, the Lord is the complete, all inspiring one. The who considers so attains completeness.
Does the Lord also need to contemplate on this idea?