Something like that would be a nice touch but I would just prefer a drinking animation, spell casting takes time and leaves you open for attack so why not potion drinking. It would also mean that the restoration skill may be a better alternative, restorative spells could take a lot of magika though. Of course this is another idea to add to a list of ideas that won't be implemented if they haven't already. It just makes a nice discussion piece.
Perhaps making potions more scarce would increase the relevance of alchemy also. If they are scarce then they would be expensive meaning it encourages the player to hunt/explore or carry out sidequests in order to prepare for more challenging quests or areas.