Clarification: Adjacent, orthogonal and diagonal term usage
Logically on a square grid there are 8 squares that can be considered as adjacent to any square, four that are orthogonally adjacent (North, East, South, West) and four that are diagonally adjacent (NE, SE, SW, NW). However, the HeroQuest rule that restricts diagonal movement below means that squares that are logically diagonally adjacent can only be moved to via an orthogonally adjacent square i.e., they effectively have a square between them for the purposes of movement and so within the world of HeroQuest they are not considered to be adjacent.
From the US rules, pages 12-13
When moving, however, you cannot pass over monsters, move through walls or move diagonally.
From the US rules, page 21
Monsters may not move or attack diagonally
For clarity throughout the rules (and related discussions on this forum) the term adjacent should be understood as referring to those squares that are orthogonally adjacent only (N,E,S,W) and the word diagonal should be understood as referring to those that are diagonally adjacent (NE,SE, SW, NW) in the logical sense but are NOT adjacent in the World of HeroQuest.
From the US rules, page 13
[H] Action 1 – Attack
As a Hero, you may attack any monster that you are adjacent to. You are adjacent if you are directly to the side, front, or rear of another square. You may, however, only attack once per turn. Your attack strength depends on your attack weapon. The stronger the weapon, the greater the number of Attack Dice used. Some weapons allow you to attack diagonally or from a distance. Most weapons, however, do not.
The precise interpretation of the phrase "to attack diagonally or from a distance" implies that one or the other is true i.e. those weapons that can attack from a distance cannot attack diagonally but I'm not convinced that this precise interpretation was intended.
From the US rules, page 15
For instance, daggers and crossbows are special weapons due to their ability to hit a monster from a distance.
Some long weapons, like the staff and the longsword, allow you to attack diagonally. The attack is made and defended normally.
Relevant text from the US Armory entry under "Crossbow"
This long-range weapon gives you the attack strength of 3 combat dice. You may fire at any monster that you can "see." However, you cannot fire at a monster that is adjacent to you.
Relevant text from the US Armory entry under "Dagger"
This sharp knife gives you the attack strength of 1 combat die. A dagger can also be thrown at any monster you can "see," but is lost once it is thrown
Taking into account the above material;
1.Ranged weapons in general, and crossbows specifically, cannot be used to attack an adjacent opponent
2.Throwing weapons in general, and daggers specifically, can be used to attack an adjacent opponent but cannot be thrown at one.
3.Ranged weapons in general, and crossbows specifically, can be used to attack an opponent who is NOT in an adjacent square which includes diagonals as these are not considered adjacent
4.Throwing weapons in general, and daggers specifically, can be thrown at an opponent who is NOT in an adjacent square which includes diagonals as these are not considered adjacent
However, in my opinion the intention of the rule above that states that you cannot fire the crossbow at a monster that is adjacent to you is to maintain a clear distinction between close combat (target in adjacent square) and ranged combat (target not in adjacent square).
The diagonal “reach” weapon capability blurs the line here as it means that under certain circumstances diagonal squares can be considered as within the range of close combat.
As far as I am aware in the base game (feel free to correct me if I am wrong here) monsters are never equipped with ranged, throwing or reach weapons so there is no amendment necessary in the official rules as there is nothing that I can see that clearly states that a crossbow cannot be fired at a target in a diagonal square (with the possible exception of the bit I mentioned above)
However in later expansions and indeed anything home brewed, a situation could occur where either a monster armed with a ranged/thrown weapon is attempting to diagonally target a Hero armed with a ranged weapon OR a Hero armed with a ranged/thrown weapon is attempting to diagonally target a monster with a reach weapon and in those scenarios there is a contention.
In keeping with what in my opinion is the intent of the original rules above, I would suggest that this contention could be avoided by inserting the following rule clarification text into the relevant Quest Books
"Ranged weapons can be fired and throwing weapons thrown at a target in a diagonal square ...unless that target is armed with a reach weapon."
In thinking this through another question has arisen in my mind ... if reach weapons can target diagonal squares (which as we have previously established are not adjacent but have an intervening square) then why can't reach weapons be used to target squares that are orthogonally two squares away? ... but this may be a bridge too far at least for now.