First off, let me say I'm glad you took the time to make this. It's very difficult putting something together with the potential of it getting shot down hard. It's very nerve wracking actually. So no matter what others may say on this or future posts, learn from everything and keep going. Keep thinking of new ideas. Especially if you find it fun. I know i do.
Putting numbers into designs is a difficult thing to do as some people get caught up in the ratios. Personally, I can be like that sometimes as well. Just look at my previous post about my thoughts on Morgana or the one about potential new items. I spent 2 days on each just looking over numbers. I learned a lot from the comments on the hextech items one as well.
Hextech: https://boards.na.leagueoflegends.com/en/c/skin-champion-concepts/hB87ETqm-hextech-items
Morgana: https://boards.na.leagueoflegends.com/en/c/skin-champion-concepts/zH1Um60g-thoughts-on-potential-changes-to-morgana
Now, on to your rework. Even if he doesn't need one, it's good to always think of how to improve the current game, or just to have fun. :D When thinking of reworks as well as entirely new champions, I like to think about, lore, early concept, what style of champion s/he is, and what I want them to be. What tools does he need to do this job or the job you want him to do? And, don't feel you have to change everything. Sometimes, current mechanics work really well.
In udyr's case, he is a spirit walker shaman who communes with the spirits of Freljord itself and was trained by lee sin in the path of martial arts. Mechanics wise, he is an auto attack based "caster", jungler, split pusher, and chases down/sticks to fleeing enemies.
My thoughts:
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Armor and Magic resist shreds are particularly tricky to work around balance wise. That being said, I love the idea of him evolving as he levels, potentially gaining shred somewhere in his evolution. I know its part of the reworked Kayle but maybe a different route would work.
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A spell shield that, upon successful shield, damages the one who cast the spell is a bit unhealthy gameplay wise. Imagine dying as nocturne the second you ulted because sivir spell shielded the darkness portion of your ult.
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I like the idea of abilities coming back faster if you auto attack more but it feels like hes would be unable to do his job if all of his abilities wore out and he was 8-10 seconds away from the nearest enemy unit.
Here is a little rework I did to the kit you came up with. I'm not trying to overshadow or anything. I just thought it would inspire you a little.
• Passive: Monkey's Evolution
- Passive 1: Udyr can recast his basic abilities over a short duration for an additional effects. At levels 6/11/16 these recast abilities are augmented.
- Passive 2: Udyr gains a % bonus to movement speed and attack speed for a duration after casting an ability.
• Q: Tiger Stance -> Apex Pursuit
- Active 1: Udyr gains a large bonus in attack speed and his attacks deal bonus physical damage for a duration.
- Active 2: Udyr's next basic attack has increased range and will lunge to the target slightly knocking back enemies in the lunge path dealing physical damage based on Bonus AD. The target of the basic attack is not knocked back but is slowed for a small amount. (can auto reset)
Level 6 augment: Range is increased and cost to recast is halved.
Level 11 augment: The attack has bonus armor penetration and no longer costs mana to recast.
Level 16 augment: The attack is guaranteed to critical strike dealing 200% damage.
• W: Turtle Stance -> Withdraw
- Active 1: Udyr gains a shield for a duration and grants him bonus Armor and Magic Resist while the shield persists.
- Active 2: Udyr's next basic attack will heal him for a flat amount plus a % of his max health increased by bonus HP.
Level 6 augment: Auto attacks now reduce the CD of Turtle Stance by a small amount and the flat heal amount is increased.
Level 11 augment: The total heal amount is doubled.
Level 16 augment: Udyr gains a large amount of tenacity upon the second activation. (does not need to auto)
• E: Bear Stance -> Ursine Howl
- Active 1: Udyr gains a large amount of movement speed that is doubled while out of combat.
- Active 2: Udyr deals magic damage based on Bonus AD and stuns all units within a radius around him for a duration.
Level 6 augment: Stunned enemies have their Armor and Magic Resist reduced for a short duration.
Level 11 augment: The radius and duration of the stun is increased.
Level 16 augment: The Cooldown and mana cost of Bear Stance is reduced to half while out of combat.
• R: Phoenix Stance
- Active: Udyr gains a large amount of attack speed and basic attacks now deal % health magic damage based on Total AD in a large cone behind Udyr's target for a moderate duration. Enemies hit by the fire are slowed by a small amount for a short duration.
This was what I thought of from your idea. I hope you keep at this. It's definitely a lot of fun!!