Let me build on that answer. Let’s take as given that we’re dealing with a maxed special (8/8).
All damage-dealers do more damage as they level up. Heroes that do a % damage key off their attack stat, which goes up each time you level the hero. Less obviously, Direct Damage heroes (e.g. Natalya) and Damage Over Time hero s (e.g. Colen) also gain in power. Although the card says they do flat damage, that damage amount is linked to the character level. Troops also modify all attack damage.
No healer’s special is impacted by level, with one exception. Like damage, healers come two flavors: % Heal (e.g. Vivica) or Heal Over Time (e.g. Ares). The first restores a % of the target’s health, so their level matters, but not the caster’s (except as to how much health the caster gets back). Heal Over Time restores a fixed number of HP per round. Both effects are modified by the target’s troops. The sole exception is Delilah, whose minions’s HP is based on the caster’s HP and, therefore, level.
Likewise, no buff/debuff hero’s special is linked to level.
Specifically, Alberich has three parts to his special: mana over time, health over time, and resurrect dead allies. None of these are attacks, so once the special is maxed, his special effect is fixed.
Zeline also has a three part special: attack all foes, remove all buffs from foes, stun foes. The attack does scale with her level; the other two (which are the two most useful components) depend solely on the special’s level.
Of course you still want to level up characters even when their specials are maxed to increase their survivability and damage from tile/slash attacks. But an Alberich at 3/70 is still extremely useful, while a striker like Marjana at 3/70 may not be as good as a similar 4* at 4/70 (which uses the same ascension mats and fewer hams/feeders).