We were playing in multiplayer mode with a mod.
After passing the turn, during the opponent's turn, my main hero was on the map for a while. After entering/exiting the hero's menu, he disappeared [possibly after entering/exiting other in-game menus]. When the turn was passed back to me, he reappeared (Fig. Kiryukha took a timeout (he's standing on Utopia, but he's not there :)), the bug repeated when the opponent was checking my troops. When I killed a Loshk, Kiryukha disappeared from the map again, but the opponent could see him on his turn (on my turn, he reappeared on my map). Throughout all this, he never disappeared from the hero panel (Fig. 2, repeated bug).
And another bug is that while playing, I minimized the game to answer in ICQ, and the game window disappeared completely, and I couldn't enter the game until the opponent passed the turn (later, when I received the turn, everything went back to normal).
P.S. These cases have never happened when playing without the HD mod.