For a heated, emotional argument:
* Fiery
* Contentious
* Volatile
* Fractious
* Bitter
* Rancorous
* Explosive
For a logical, reasoned argument:
* Persuasive
* Compelling
* Convincing
* Logical
* Rational
* Cogent
For an argument that is unfair or unreasonable:
* Unfair
* Unjust
* Prejudiced
* Biased
* One-sided
* Invalid
* Flawed
For an argument that is prolonged or drawn-out:
* Protracted
* Lingering
* Endless
* Unending
* Tedious
Ultimately, the best adjective to use will depend on the specific nuance you want to convey.