An electron is an excitation of a quantum field that has a defined set of characteristics. You don't seem to be able to tell me anything at all about the what a "self" actually is. It's some vague part of our mental experience that includes memories but doesn't include them, is all I've learnt from you so far.
You're trying to tell me that something that you can't actually define at all or tell me where its boundaries lie, is "fundamental".
I'm just asking what it is - not what it's made of or how it works, just what part of people's minds do you think is the "self" or consciousness.
Consciousness is what you are born with. As a new born infant we don't have a mind or self awareness or Ego. All these things are built later. Consciousness or what we can call Unconscious processes, direct the development of the baby, building up its conscious mind, self awareness and other mental processes.
Consciousness is what remains when the mind is absent. This happens at the time of birth, can happen through dementia or even during meditative practices.
Consciousness is not just what we normally think of as wakefulness or the conscious mind that we are aware of. Consciousness consists of deeper layers that are normally called the Unconscious mind. It is generally believed by philosophers that Consciousness is like an iceberg. A major part of it lies below the surface and is more powerful than the conscious mind. This is the Unconscious mind.
It is generally now understood that our decisions are taken by unconscious processes before the conscious mind is even aware of it.
Coming to your question...the Self can be considered as the Unconscious mind that lurks behind the conscious mind and directs it. In other words, the conscious mind is just a projection of the Unconscious mind or the Self.