Is there a "good" consciousness?

Consciousness is like a cup which is holding water. Without consciousness you will not exist. Buddhism talks about four elements and some relgions three elements, and some five as the basic building block of various objects of Nature including humans. Consciousness is the holding structure for these elements. But if you see it objectively it is just a container which is holding you including mind, various sensory organs etc etc. Can you label a container good or bad? You have trillions of cells organized inside your body. What the is organizing principle here?

Nirvana is a realization that consciousness is not the final thing. Infact it is not a thing. Nirvana does not break consciousness otherwise you will die. It is just that consciousness will not manifest again as a body. Upon death it will be merged with the whole or Sunya or nothingness. Your desire gives a shape to your consciousness. For example if you desire to be a scientist in this birth and it is not fulfilled, consciousness will give it a shape in next birth. You might have PhD parents in your next birth.

Nirvana is a direct experience of something beyond consciousness. Call it superconsciousness or Sunya. From that point on more desire and hence no more movement of consciousness.

It is the most difficult concept intellectually to grasp.

I wondered if there was a "good" (I mean that in a non-technical way) consciousness?

Consciousness exists as a taker of objects. Consciousness works together with perception to cognize an object in the cognitive process.

Consciousness is not a thing. The problem is that we are forced to talk about it in conventional terms using nouns. That can give the impression that there is "something" there but there is not.

Since there is nothing there, there is nothing to superimpose qualities onto such as "good, bad etc.". The act of superimposing qualities such as "good", "bad" etc. onto objects are taught by the Dalai Lama in his book "How to see Yourself As You Really Are" in the chapter "Identifying Ignorance" on p. 45-49 and the chapter "Why Understanding the Truth Is Needed" on p. 63 where he writes:

"The supreme scholar-practicioners i India - Nagarjuna, Aryadeva, Chandrakirti, and Dharmakirti - understood that the truth cannot be realized without seeing that we superimpose on people and things a status of solidity and permanence that actually is not there. The emptiness of that false superimposition must be understood, and to do this they analyzed phenomena through scripture and through reasoning".

And further:

"When ignorance is overcome, you will have uprooted the mistaken beliefs that superimpose on objects qualities such as beauty and ugliness beyond what they actually have".

I have made a highlights in this above sections.

What sort of cognition is the cognition of extinction?

I am not sure i understand your question correctly. I will try to answer as best as i can.

When dealing with conditioned phenomena both physical and mental they all follow a specific process. That process is "arising, presence, dissolution".

When doing insight meditation one can observe this process clearly. The dissolution/extinction part of phenomena happens e.g. when the abdomen has reached full extension on the inbreath and then begins to fall. That is the start of the dissolution step. When the falling of the abdomen has finished then the dissolution step has finished and one can clearly see the impermanence of the breath. This dissolution step can be seen in all phenomena and when observed it will grant access to understanding the impermanent nature of conditioned phenomena.

This is taught by Ven. Mahasi Sayadaw in his book "A Discourse on the Anattalakkhana Sutta" in the chapters "Refuting A Self Apart From the Five Aggregates" and "Why It Is Called Impermanent" both on p. 85-90.

Lanka

What is Good? We usually see a person who does not cause others harm and is basically of an open benificent attitude "good." Anything can be described as good based on how open and non-harming it is IMO. Similarly, a state of consciousness which is Good... There is a mundane level and super-mundane level to this idea. The mundane level is this: metta aka loving-kindness: being in an overall attitude of unconditional loving-kindness as well as the other brahmaviharas, mudita, karuna, upekkha.

The supermundane level is probably what you were asking about... Consciousnesses are only a part of the True Consciousness. They are just imitations and second-level projections.

The best way to reach this True Consciousness is to naturally let it appear by never trying to reach it in the first place.

So to answer your question, yes, the best consciousness is the consciousness in which you are free from suffering and dualism. It is also uncreated and unborn (as the sutras state). Thus, to achieve the best level of consciousness, let go of all levels of manufactured consciousnesses/ideations (including the idea of achieving itself) and only the truest can remain.