Alvaro has the right of it for true turning costs, though I think he may just want to prevent zigzagging, so you could alter the F cost to make it more expensive to move to a node if it involves moving in a different direction than the last move. Maybe store the dx/dy for each move, then compare it to the new potential, and roll it into the F cost?

No, if nodes represent locations they "forget" the direction of the last move. The cost of leaving the node is the same for every path through it. Only nodes representing position and direction can distinguish turning from straight movement.

Example with no obstacles:

a..
41.
752
.63
..b

As the preferred path from a to b, you want 1-2-3 (diagonal, diagonal, straight, straight) or its symmetrical counterpart 4-7-6 (straight, straight, diagonal, diagonal) rather than 4-5-6 (straight, diagonal, straight, diagonal); the status of 1-5-6 (diagonal, straight, straight, diagonal) and 4-5-3 (straight, diagonal, diagonal, straight)

is unclear. At node 5 you want to prefer 3 as next node if coming from 4, 6 if coming from 1, demonstrating the need to split location-only nodes into location + direction nodes.

Unfortunately the cost of all four paths is going to be the identical with any fudging of diagonal step cost away from the correct factor of sqrt(2), because all have two straight steps and two diagonal steps.

Fudging costs can only affect search order, which doesn't seem a very good idea if you like symmetry and consistent behaviour.