The looking that I suggest, which I got from John Sherman, is often confused with self-inquiry. I'm guilty of making that confusion myself, and sometimes in my writing I will use the terms interchangeably. However, that is really just me being lazy.
The term "self-inquiry" doesn't fit for a couple of reasons. First, self-inquiry often carries the connotation of being a therapeutic device. It is used to dig out things like habit, assumptions, and denied thoughts. All of that is in the "how to be" realm, and not in the "what you are" realm. It's a way to examine what you do, and why, in order to clarify or change your behaviors, or alleviate suffering. All of that is good, of course. It's just not what the looking is about.
Secondly, even when self-inquiry is a suggestion within the "what you are" realm, it usually comes in the form of a question, or in the flavor of an examination. These are not the looking because the looking is nothing more than a turning of attention directly on to you. The point is not to answer a question, or make a discovery. It's not done in order to get new information. The looking is only looking, and it's that simple looking which exposes the lies of the separate self to the truth of the self.
The looking is not a therapeutic device, even though it can have a therapeutic effect. It can be accompanied with a rush of clarity, or insight, or openings of consciousness. Those are all beside the point, and if they distract you then that's what they are, distractions. They are not bad, they just aren't the simple looking. The looking usually occurs in brief moments. What it reveals is utter simplicity itself. The brain is not built to deal with the simple. The brain looks for categories, complexity, motion. That's why it's easy for the brain to get sidetracked into making something out of the looking. Don't be fooled. The looking is only looking.