Ok, so in my most recent session, one of my players ended up dying. No major issue there, they tried to do something stupid and failed.  Now he wants to role up his new character as a Kensei Monk with a Longbow, again I have no problem with this. What I do have a problem with is understanding how he wants to use it in melee combat should he find himself surrounded. 

According to PHB under the "improvised weapons" section, if you use a ranged weapon to make a melee attack, the damage dice is 1d4 but since the Longbow is his Kensei weapon (hence, a monk weapon), would the damage dice equal his martial arts dice (1d6)?

P.s. I'm aware he can make unarmed attacks whilst holding the longbow, thus bypassing this issue. I just want to know if its RAW.

I would say no the longbow lose all property once it used as improvised weapon unless the GM rule otherwise.

Improvised weapons are intentionally lax in their classification. Anything that isn't an actual weapon, or is a weapon that isn't used properly, can fall in the category. However, if the end result resembles a weapon, it can be treated as that weapon, including relevant proficiencies (like the example of the book about a table leg being used as a club).

Going to the monk, there's no specific weapon trait that signifies it as a "monk weapon". It's a classification of weapons with specific traits mentioned in the monk's "Martial Arts" trait. More specifically, "shortswords, and any simple weapon that lacks the two-handed or heavy property".

So, if the longbow is unstrung and used as a club or staff (or anything else. Perhaps the bow has a pointy tip on one end and it can be used as a spear when unstrung), check that weapon against the monk weapon classification (all three examples here count, for instance).

If the longbow is just swung about I suspect it resembles no particular weapon, thus it's an "improvised weapon" (not a simple weapon) that deals 1d4 damage and has no properties. Since it doesn't fit the classification for monk weapons (or Kensei weapons), it can't be used with Martial Arts.

Agreed. A longbow used in melee is no longer a longbow, it's just an awkward stick. The player is being creative though, so maybe offer them a way to make this dream a reality: let them choose longbow and quarterstaff or glaive as their kensei weapons and have it be known that the character is searching for a lost kensei monastery where they make and wield magic bow-staves/glaives. If the monk earns it in the next few sessions by finding the monastery you just invented then give him a minor magic weapon of a longbow which can be converted to a staff/glaive (or back) using the player's free object interaction on a turn. The gnomish monk/inventor at the monastery also had created a more powerful weapon in this style which was stolen by [insert some bad guy here] and they would appreciate it being wielded instead by someone faithful to the ways of Kensei. This will give the player the Green Arrow/Star Wars Rogue One Blind Monk aesthetic they seem to be going for, while also hopefully giving a sense of earning it, and avoid the nonsense that is improvised weapons.

If this is AL, then yeah follow RAW and it would be an improvised weapon dealing 1d4 damage and unable to use martial arts.  If this is a Homebrew game/world, I would make him take the Quarterstaff and Longbow as his Kensei Weapons at 3rd and let his "Monastic Order" wield a specific unique weapon that is a combination of the two and can switch from a ranged to melee weapon that incorporates the following:

• Used in melee, it is a Quarterstaff that can use Kensei abilities and use Monk flurry of blows, Martial Arts and so on. 1d6 one handed. 1d8 versatile
• If used as a ranged weapon it can function as a Longbow, 1d8 ranged.
• Make switching the weapon, (stringing/Unstringing) either a bonus action or reaction for the player, or perhaps cost a KI point, or some other cost associated with the switching.  This will give the player the character fantasy he wants but he has to pay for it in some fashion.  
• He most likely will never find a magical version of this weapon, since it was created specifically by monks for monks of his "order", it's a non-issue for a Kensei after 6th, but he may feel differently if the other players get some fancy weapons with neat abilities.  If he ever lost it or it was destroyed he would be forced to return to his Order and attempt to get a new one.
• (RA Salavatore's ranger Elbryan Wynden in the DemonWars Saga had a weapon exactly like this)

Alright, so the general consensus is that Kensei Longbow in melee is 1d4, regardless of martial arts? Thanks for the clarification.

I'll let my guy know he should stick to unarmed strikes while in melee range.

I came up on this randomly but this is such a weird discussion. If your player can deal the same damage by just using the unarmed attack anyway without incurring any penalties or unequipping the longbow then why is it an issue to just let him "attack with the longbow" and do the same damage. You're just Imposing a restriction that is entirely down to flavor and has no eventual effect on mechanics at all. Nonsense, just let him flavor the attack as attacking in melee with the bow, it makes straight up no difference besides being overly restrictive.

The difference is that if still treated as a longbow Monk Weapon when attacking with it as an improvised weapon, then the Kensei features may apply.

If the player is trying to metagame and eek out a few extra damage then you tell the player to get a grip. If the player just wants the flavor of attacking with his bow make it mechanically an unarmed attack and that's the end of that. I'm still not seeing the point of this discussion

Agile Parry, 3rd level, requires you be wielding a Kensei weapon and melee. If the longbow is being used as an improvised weapon in melee, does it count and give you the +2 AC? Is it a Kensei weapon at that point? If not can use your martial arts die then it can give you the AC, which your “flavored” unarmed strike as a longbow melee attack cannot.