AP Hydra
An AP Hydra is no new suggestion, but I was unable to find any attempts at building the idea and justifying it. In turn, here's my attempt at building the idea and justifying it based on the current state of the game.
The shape of the AoE is probably the most debatable question, since there are pros and cons to each. As a tiebreaker, I feel the cone of 
is ideal for and unusual reason: I tend to consider 
the straightforward upgrade to Tiamat, both being AD-focused, and the lifesteal being merely secondary; therefore the tangent upgrades, and this, get the cone.
Next would be the recipe. Now, the recipes of the other Hydras is Tiamat and two other components that contain the stat in which the Hydra scales with, one being an advanced item having a second stat or side effect, and then a combine cost that gives the Hydra a total value of 3500. That said, the AP Hydra would then be built from Tiamat and two AP items, one being an advanced item with second stat and/or side effect. I feel 
is the best fit as the advanced item, as the other hydras offer health or lifesteal to increase sustain, and it also compliments use of 
in recipe. That gives us a working recipe and raw stats.
The Cleave/Crescent damage is probably the trickiest, though the easy route is probably making Cleave deal the same damage as Nashor's Tooth, and not have any on-hit magic damage to the target. Crescent more than doubles the cone damage and also applies the its bonus damage to the target.
+ + 
+ 665 = 3500
+25 AD (I didn't feel this upgrade would increase Tiamat's AD) +75 AP (+20 gained from upgrade) +45 armor
CLEAVE - Basic attacks damage enemies in a cone behind the target 15 (+15% AP) magic damage. CRESCENT - Your next basic attack deals an additional 20 (+20% AP) magic damage to the target and enemies in the cone (does not increase range). (20 second cooldown). CRESCENT resets your autoattack timer.
Seeing as the cone deals a vast majority of the damage, I feel it compliments the other Hydra's well, focusing more on wave clear than champion damage, but also offering stats to enable that through abilities.
