Shatar is indeed an uncomplicated checkmate game, just marginally more complicated than chess . Shogi, I think, is at the other end of the spectrum in terms of complexity: many pieces with asymetrical move patterns and the possibilites of drops.
The idiosyncratic rules of Shatar
Despite the funny checkmate rules, I would consider the game slightly simpler than orthodox Chess. Because of the less-powerful Queen, and absence of the special moves (castling. double-push, e.p. capture, under-promotion). How exactly you execute the checkmate has very little influence during most of the game. (You should of course stay aware of which material combinations have no mating potential, which is different than in Chess, but not more or less complex.)
The only other games I am aware of that have restrictions on how you can checkmate are Shogi variants: you are not allowed to checkmate there by dropping a Pawn. I never understood why they adopted that rule; it doesn't seem very common that you can checkmate through a Pawn drop.
Games with drops are indeed much more complex than the corresponding game without drops. Crazyhouse (orthodox Chess with drops) might be even more complex than Shogi, despite the smaller board (8x8 instead of 9x9). This because the pieces are on the average much more powerful.
There also exist western Chess variants without drops that are very complex, due to a large board and a large number of (very poweful) pieces.

well, I find it interesting. ty
the older game had a very limited bishop (called the elephant)- so I would NOT have predicted that we see an innovation on the bishop's movement from Mongolia.
OTOH; I have played the older game and with a weak queen, and a lesser bishop; it CAN be a pretty slow game.
Shatar was a checkmate game played for many centuries in Mongolia, before it was replaced by FIDE chess by pressure of the former USSR. This game is a direct offspring of the medieval Arabic chess, evolving in a different direction as chess did in Western Europe and Russia. The movement of the queen is particularly interesting, as it has the combined moves of a Ferz (general) and Rook.
In 1993, AISE organized a `Grand Prix' tournament (played by postal mail) on Shatar. In later years, more Shatar tournaments were held in this game.
Rules
The same pieces and the same startup position as orthodox chess are used, but with the following modifications:
Rooks (Terge) and Bishops (Teme) have the same moves as in orthodox chess.
The first move of the game is obligatory: 1. d2-d4, followed by 1. d7-d5.
There are different types of check: Shak is given by queen, rook or knight; Tuk is given by a bishop, and zod is given by a pawn. As written above, a mate with a knight is forbidden. Moreover, one may not mate except by a shak (i.e., checking by queen or rook), or by a mate that is preceded by a series of checks that included at least one shak.E.g., if we mate the opponent after checking with a knight ( a shak ), then in the next move with a pawn, and then mating with a bishop, then the game is won. This rule means that as far as sacrifices are concerned, you must reserve at least a knight, a rook or a queen to fulful the requirement of a shak if you have any intention of winning the game. If one mates the king without fulfilling the criterium, i.e., with a bishop or pawn, or after a series of bishop and pawn checks, then the game is a draw - this is called Niol. Also, if a player has only a king and no other pieces left, then the game is a draw - this is called Robado.