Bismillahi Al-Rahmani Al-Rahim

A Fifth Maddhab

assalaamu `alaykum w rahmatullahi wa barakaatuh

Dear brother

Is there a fifth madhhab in addition to the madhhahib of Imams Abu Hanifa, Malik, Al-Shafi`i and Ahmad?

Can one, for example, follow the madhhab of Twelver Shi`a?

What is the difference between the Imams of Ahl Al-Sunnah and the Imams according to the Twelvers?

wa `alaykum al-salam wa rahmatullahi wa barakatuhu

As for the first question

Is there a fifth madhhab in addition to the madhhahib of Imams Abu Hanifa, Malik, Al-Shafi`i and Ahmad

the brief answer is no, there is no fifth madhhab in addition to the four madhahib of Ahl Al-Sunna, namely the madhhahib founded by the four scholars you mentioned above (may Allah be pleased with them). Ibn Rajab (may Allah be merciful with him) mentioned the basic reasons for this in his rebuttal against those who do not follow the four madhhahib in several places. But first a bit of context
[The preservation of fiqh through four schools] is among Allah's gentle kindness toward His believing slaves, and among the mass of His customary laws in preserving this religion. If not for this, people would have seen the wonder of wonders: every imbecile, hot-headed, insolent, sham who was impressed with his opinion would claim that he is the greatest of all Imams, that he is the one to guide the umma, that people should turn only to him, and that no one else should be relied upon.
However, with praise for Allah and His gracious bestowal, this door of great danger and tremendous weight was barred. And this great corruption was constrained. This is among gentle kindness, beautiful habits, and tender mercies of Allah Most High towards His slaves.

In spite of this, people still appear, claiming to have reached the level of ijtihad, speak about knowledge without restraint, or following one of those Imams. This is tolerated from some of them because of the outward veracity of their claims; for others, their statement is rebutted and they are declared liars in their claim. As for everyone else who has not reached this level: they are only capable of following one of these Imams and following what the rest of the umma did. (Ibn Rajab, Al-Radd `Ala Man Ittab`a Ghayr Al-Madhahib Al-Arba`. Unpublished translation, copyright Musa Furber 2002 CE. p7-8)
A nd now Ibn Rajab's words regarding this question can be better understood
If it is said, We concede to preventing the general masses from following the way of ijtihad since this leads to the greatest of wrongs. /34/ However, we do not concede to preventing following an imitated, mujtahid Imam, outside of those famous Imams.
It is said: We have pointed out the reasons for preventing this: it is that the other unpopular schools have not been corrected and perhaps something is attributed to them that they did not say, or understood that they did not intend. Their schools have no one to defend them and point out the mistakes that took place in them, contrary to these famous schools.
If it is asked, So what do you say about another Imam's school if it has been recorded, corrected, and memorized, just like their schools?
It is said: Firstly, this is not known to exist right now. If we assume that it did take place right now, and conceded the permissibility of following it and affiliating with it, this would not be permissible except for someone who visibly shows affiliation to it, gives verdicts according to it, and defends his school.
(Ibid, p8-9)
F or further clarification, Sheikh Nuh Keller mentions in the second introduction to The Reliance Of The Traveler quoting `Abd Al-Rahman Ba`alawi
Ibn Salah reports that there is scholarly consensus on its [sic] being unlawful to follow rulings from schools other than those of the four Imams, meaning in one's personal works, let alone give court verdicts or formal legal opinions to people form the, because of the untrustworthiness of the ascription of such rulings to the scholars who reportedly gave them, there being no channels of transmission which obviate the possibility of textual corruption and spurious substitutions.
The Zaydis, for example, who trace themselves to Zayd ibn Husayn (n: son of `Ali and Fatima), the beatitude of Allah be upon them, despite the fact that Zayd was one of the Imams of the religion and a renowned figure well qualified to give guidance to those seeking it, his followers identify him with extreme permissiveness on many questions, ascriptions based on failure to check as to what his positions actually were (n: by naming the intermediate transmitters and establishing their reliability). It is quite otherwise with the four schools, whose Imams (Allah reward them) have spent themselves in checking the positions of their schools, explaining what could be rigorously authenticated as the position of the person it was attributed to, and what could not be. Their scholars have thus achieved safety from textual corruption and have been able to discern the genuine from the poorly authenticated. (Bughya Al-Mustarshidin Fi Talkhis Fatawa Ba`d Al-'Aimma Min Al-Muta'akhkhirin, p8, via Reliance Of The Traveller, b7.6)
S o the answer is that no, there is no madhhab in addition to the four madhhahib of Ahl Al-Sunnah that is permissible for Muslims to follow.

As for the second question

Can one, for example, follow the madhhab of Twelver Shi`a?

Based on the above the answer is, quite clearly, no. In addition to the problem of lack of tawatir [such large numbers of people in each generation transmitting the madhhab such that it is statistically impossible for them to have simultaneously agreed upon a lie] there is the issue that what the Twelvers follow is not a single madhhab. The Twelvers try to confuse Muslims into thinking that the four madhhahib of Ahl Al-Sunnah are proof of division and misguidance--in contrast to their one single madhhab which they claim is proof of their unification and pure guidance. This is little more than wishful thinking on their part. There is no single Twelver madhhab, and how could there given their system of mutjahids and referent scholars? And even if this were not the case, how could they maintain that the fiqh they have today is historically connected to their founding scholars given the bizarre paradigm shift in usul that they underwent in the later centuries? And how can they claim this when the `aqidah of their major scholars keeps changing on major issues?

So even once again the answer is quite emphatically in the negative.

And as for the third and final question

What is the difference between the Imams of Ahl Al-Sunnah and the Imams according to the Twelvers?

the differences are many, but perhaps the greatest differences are that we do not declare a single one of these Imams to have hidden knowledge or be of higher rank than the Prophets (Allah bless them all) and we do deny that they are divinely protected. We do not make belief in them or acceptance of a single one of them a condition for sound belief. We also do not claim that what the Prophet (Allah bless him and give him peace) was incomplete and in need of the Imams of Ahl Al-Sunnah to complete it, and we don't give the Imams authority to abrogate or invent. Imam Al-Dhahabi in the beginning of Al-Muntaqa gives some basic guidelines as to the the differences, namely that we don't require anyone to believe in them and we don't say that things happen because of them nor through them: we don't believe in them as the [Papist] Christians belive in their Pope (Al-Dhahabi. Al-Muntaqa Min Mizan Al-`Itidal. p7-8). But then again, none of the Imams of Ahl Al-Bayt (peace be upon them) made these claims for themselves.

wa al-salamu `alaykum