The words "Alpha and Omega," the first and the last, is too small of a definition of God because it is limiting. "First and Last" suggests that there was a beginning and there will be an end of God. That is not true! Even though my natural mind cannot understand, I believe that God always was and always will be. He is the "I AM" or "Everlasting" God. He is El Olam - everlasting, never ending, the Eternal God. He inhabits eternity in the past and in the future. Doesn't those two words, alpha and omega, contradict the true nature of "I AM" by implying that God had a beginning and an ending?
The same seems to be implied in "Ancient of Days" because this suggests that a time element is linked to God as if there is an amount of days past - ancient - that identifies when God first began. It is an "ancient number of days." God is not caught up in time as we know it on earth. There are no hours, days, or years with God. It is a term we use in the natural. Yet, how can we describe God without using our natural form of speaking and in our experiences of time? Yes, God is the Ancient of Days - but He is much more than that! He always was, is and always will be. There was no beginning as the phrase "ancient of days" seems to imply.
I like the title "I AM who I AM" because no amount of English words will ever describe completely who God is. English words will always be too limiting for who God, Yahweh, completely is. He is the Great I AM. He is who He says He Is. By faith I believe that!