What is the invisible consciousness or consciousness without surface? From MN 49 (trans. Sujato):


  Consciousness that is invisible, infinite, radiant all round—that’s
  what is not within the scope of experience based on earth, water,
  fire, air, creatures, gods, the Creator, Brahmā, the gods of streaming
  radiance, the gods replete with glory, the gods of abundant fruit, the
  Overlord, and the all.
  
  Viññāṇaṃ anidassanaṃ anantaṃ sabbato pabhaṃ ...


The invisible consciousness (viññāṇaṃ anidassanaṃ) from MN 49 seems to be different from the sense consciousness described in MN 18 e.g. eye-consciousness, ear-consciousness etc. because it is not within the scope of sense experience, according to MN 49. It is also found in DN 11. It is also translated as "consciousness without surface" (from here, trans. Thanissaro).

Question 1:
What is the invisible consciousness or consciousness without surface described in MN 49 and DN 11?

Is it a cosmic consciousness like the one found in Hinduism? I guess not.

Question 2:
This answer implies that this invisible consciousness or consciousness without surface is the re-linking consciousness (patisandhi-viññana) that connects one lifetime to the next. This also implies that the re-linking consciousness (patisandhi-viññana) which is the invisible consciousness or consciousness without surface (viññāṇaṃ anidassanaṃ) is the consciousness that descends into the womb causing materiality-mentality (nāmarūpa) to arise (from DN 15). Is this right?

Question 3:
Also, the description of "invisible, infinite and radiant all round" sounds very similar to the "luminous mind" (pabhassara citta) from AN 1.51-52. The same word "pabham" (luminosity) or "pabhassara" (luminous) is used in both MN 49 and AN 1.51-52. How is the invisible consciousness or consciousness without surface related to the luminous mind?
Few thoughts, based on my understanding of Buddhism:

One: Vijñāna is not an entity, not a substance - it is an emergent effect, an emergent phenomenon known in modern terms as "subjective experience" or "subjectivity". Most people, when they read this phrase ("Consciousness that is invisible, infinite" etc.), assume it means consciousness as something objective that exists or dwells - like electromagnetic field - but in Buddhism it means subjective awareness, subjective experience, the continuously unfolding content of cognition. So we're not talking about cosmic consciousness that lives in space, we are talking about one's actual state of mind.

Two: on many occasions Buddha spoke about "support" and "condition" and "ground" - and how any such support/condition/ground is a basis of dukkha. He then advocated transcending any notion of ground, to achieve a dynamic state of mind which is ground-less, and therefore unconditional. Consider the following image from SN 12.64: 


  "... Just as if there were a roofed house or a roofed hall having windows on the north, the south, or the east. When the sun rises, and a ray has entered by way of the window, where does it land?"
  "On the western wall, lord."
  "And if there is no western wall, where does it land?"
  "On the ground, lord."
  "And if there is no ground, where does it land?"
  "On the water, lord."
  "And if there is no water, where does it land?"
  "It does not land, lord."
  "In the same way, when there is no [ground such as] desire, attachment, craving, then viññāṇaṃ [=cognition or cognized] does not land and does not grow."


Also, in MN21 there is a following image:


  "... Suppose a person was to come along with dye such as red lac, turmeric, indigo, or rose madder, and say: ‘I shall draw pictures on the sky, making pictures appear there.’ What do you think, mendicants? Could that person draw pictures on the sky?”
  “No, sir."
  "Why is that? Because the sky is formless and anidassano. It’s not easy to draw pictures there."


Here the word is used in context of blame and blamelessness, indicating a condition when one's pure morals provide no ground for any blame. So, anidassano refers to something that provides no surface to lean on, no basis for something to happen or to be done.

Adding points One and Two together, we can reach conclusion that Viññāṇaṃ anidassanaṃ means subjective experience that is completely open and groundless, without support, that feels like the infinite radiant sky.

This of course matches all of my teachers' descriptions of Enlightenment. For official sources, here's Dogen:


  The water is clean, right down to the ground,
  Fishes are swimming like fishes.
  The sky is wide, clear through to the heavens,
  And birds are flying like birds.


or Pema Chodron, a student of Chogyam Trungpa Rinpoche:


  "...Then he [Trungpa Rinpoche] goes on and he talks about the mantra. And the mantra is: OM GATE GATE PARAGATE PARASAMGATE BODHI SVAHA.  
  
  In other words, a way to practice the profound prajnaparamita is actually to say this mantra —as well as the on-going practice of continually letting go, or letting be, training in a flexible, open, ready mind. 
  
  [Chogyam Trungpa] Rinpoche's translation is: OM, GONE (GATE is gone), GONE, (then PARAGATE) GONE BEYOND, (PARASAMGATE) GONE COMPLETELY BEYOND, (BODHI) AWAKE, (SVAHA) SO BE IT. So: OM, GONE, GONE, GONE BEYOND, GONE COMPLETELY BEYOND, AWAKE, SO BE IT.  
  
  There's lots of translations of this, and one is: OM, TRANSCENDING, EVER TRANSCENDING, TRANSCENDING EVEN TRANSCENDING, TRANSCENDING EVEN TRANSCENDING OF TRANSCENDING, SUCHNESS, SO BE IT. 
  
  What is wonderful about this mantra is that it is not a description of some fruition. It's actually a description of a journey that we are all on. We are all on this journey of going, going, going beyond, going even beyond.  
  
  No matter where we are, we can move on to the next beyond. Do you see? It's not a description of: I made it! It's like this! It's a description of: OM, groundless, even more groundless, can it get more groundless than this, Oh my gosh, it's ultimately groundless, there's no ground!, and then BODHI could be translated as Aaaaaaaaaahhhhiiiiiiiii! [Makes a falling scream] So be it! [Audience laughs] 


To summarize, I think


  Viññāṇaṃ anidassanaṃ anantaṃ sabbato pabhaṃ


means 


  Cognition with no support, unbounded, completely transparent.


or perhaps, if we take viññāṇaṃ with an -m to be an adjective, then 


  Providing no ground for the cognized [to land and grow into dukkha], without a limit [to stop it], completely transparent.


which reminds of the following instruction given by Buddha in Bahiya Sutta:


  "... Then, Bāhiya, you should train yourself thus: In reference to the seen, there will be only the seen. In reference to the heard, only the heard. In reference to the sensed, only the sensed. In reference to the cognized, only the cognized. That is how you should train yourself. When for you there will be only the seen in reference to the seen, only the heard in reference to the heard, only the sensed in reference to the sensed, only the cognized in reference to the cognized, then, Bāhiya, there is no you in connection with that. When there is no you in connection with that, there is no you there. When there is no you there, you are neither here nor yonder nor between the two. This, just this, is the end of stress."


Either way this is clearly a description of Non-abiding Nirvana (apratisthita-nirvana), also known as "suchness" (tathata).