Well if you want a slightly different take, when we did MageKnight Dungeons, we decided to keep the rules simple and consistent.
Turning had no cost, but it took 1 Movement Point (MP) to more forward a square unless that square had a number in it. That number represented how many MP's it took to enter that square. A figure could always move one square on it's turn no matter how many MP's it had left unless it started the turn with 0 MP's.
To move diagonally, it cost +1MP above normal movement. Ie. if it would normally take only 1 MP to move into the square, to move into it diagonally, it now took 2. If the square normally took 4 MP's to move into, it now took 5 MP's do move into it diagonally.
The only typical question that arises when attempting to move diagonally using the 1-2-1-2 etc is;
Does it matter from turn to turn if one continues to move diagonally?
Ie. a figure moves 3 squares diagonally 1-2-1 for a total of 4 MP's. Next turn does it cost the figure 2 MP's to continue moving diagonally?
I ask because if you are going to get realistic, the answer is yes. If you are trying to keep the night mare
of tracking things low, the answer is no.
Or you can make each diagonally move the same cost no matter what.