
Yes, it sounds like you may have dissociative states.

There are various names to this (all under Dissociative Disorders) as ego states, alters, alternate personalities etc. It doesn't matter what you settle on (or, if they are aspects of yourself, they will help you decide what to call it) as long as you are in treatment for a DD.
If they are "alters" as you question, they might have their own "real" names. As the communication between your parts increases, and as those parts of you feel safe, they will share names and other information with other parts.
Fear not! Since all of this conversation is inside your mind, it's not schizophrenia or schizoaffective disorder, I'm pretty sure.

AND each aspect/part is YOU and only YOU.
The main difference between multiples and singletons is that multiples (those with DID) have a lack of "continuity" between thoughts and desires. Another words, like when someone goes to a fun place (Disney, zoo, ball game) they may allow their "inner child" to rule and make some decisions (cotton candy, hot dogs, fun ride etc). The dissociative person might have a certain part or aspect of them that wants and does this, and the other parts may not remember it. The singleton does remember, and in fact never loses time -dissociates- where the multiple does.
As a dissociative person begins to heal, that continuity increases, the gap of information decreases.
But, no matter where you are on the continuum of dissociation, don't worry, it's ALL YOU!
I don't know where you are, but most psychiatrists don't treat DD/DID. A psychologist with experience with dissociative disorders is what you will want, imo.
PS (Hope this isn't too much information) The fact that you are now able to discern the dissociated times and admit to the conversations you are having with yourself, means that you are ready to begin to progress and heal. Dissociation works quite well for a while --sometimes for years-- but as a person ages, the usefulness breaks down and in fact, may begin to cause problems.