I'd be surprised if it needed an AGM battery if it's not the ISG/START STOP model. I'd also expect the main dealers to have highlighted if the battery voltage was low. Do you happen to have a multimeter so you could check the voltage yourself?
A DVM wont tell the full story, you need a load tester.
A 20 second google found this, i would get a new battery as the one in the photo is no where near this spec, these are all 95Ah- assuming its not the 74-77Ah Type that is used (it seems to have two variants)
Hyundai ix35 - diesel engines:
2.0 CRDi (184 KM)
BRAND AND SERIES MODEL CAPACITY COLD CRANKING AMPS DIMENSIONS BASE HOLD-DOWN PRICE
Bosch S4 S40 28 95 Ah 830 A 306x173x225 mm B01 More
Exide Premium EA954 95 Ah 800 A 306x173x222 mm Korean B1 More
Exide Excell EB954 95 Ah 720 A 306x173x222 mm Korean B1 More
Varta Blue Dynamic G7 95 Ah 830 A 306x173x225 mm B01 More
These diesel engines should work with batteries of dimensions 306x173x222 mm or 306x173x225 mm, which capacity is minimum 92 Ah. The most basic battery of this set is Exide Excell EB954 of a capacity 95Ah and 720A Cold Cranking Amps, the battery with the biggest capacity is Varta Blue Dynamic G7 of a capacity 95 Ah, having 830A Cold Cranking Amps.