Do modern submarines have an artificial horizon on the steermans board?

Answer 1

No - in the 70's, there was a monitor system called the Conalog System, which sensed the boat's movement, depth, bubble and course and displayed it on a monitor that sat in front of the planesman and gave him a 3D line display that appeared to move toward him, or in the direction he was steering. It was eventually replaced with standard gauges (depth, rudder, bubble, etc.) because it tended to cause the planesman to become somewhat fixated on the display, even to the point it would cause them much difficulty in staying awake. "Hypnotize" and "Trance" aren't really accurate, but they're pretty close as to what was occurring.

The newer class submarines may use electronic display systems for steering, but if they do, they're classified. Any electronic display would likely be an electronic repeater of a mechanical input anyway, since the Navy typically doesn't go back to a system that they've trashed previously because of problems without good reason to believe it won't happen again. Critical systems tend to be mechanical in nature (even if they're just backups), since most major casualties will usually result in an electrical failure or emergency shutdown of a Main Electrical Bus, which could render any electronics useless.

Having gauges to look at might seem boring, but in reality they keep you more alert. It takes skill to maintain proper bubble, course and depth while submerged, and keeping it harder for the planesman & helmsman to do their job makes them more alert and responsive.