Why do some strands of Islamic terrorism wish to 'purify' the world?

Islamic Terrorism, by definition, are the terrorist acts committed by groups who profess Islamic or Islamist motivations or goals. The ideology lying behind some of the radical or extremist views and actions are taken from the Islamic perspective of the Quran and the Hadith. Islamic Terrorism, under groups such as Al-Qaeda and Islamic State, can be categorized as 'global terrorism' as it aims to degrade and damage the unipolar global power, the United States, or the West itself. However, due to the nature of this radical and religion based movement, could be considered 'new terrorism', or in it's attempt to overthrow the colonial rule and imperialistic nature of US occupation in the Middle East, 'nationalist terrorism'. 

It is understood that a large part of the Islamist agenda is the objective of 'purification' of the world, this stems mostly from the idea of 'Jihad', or 'the struggle'. This struggle can address the 'greater' struggle, internally, which focuses with one's personal and psychological acceptance of their own Islamic faith, while the 'lesser' struggle refers to the physical barriers to Jihad, which enforces the acceptance of faith and conversion of others to a like-minded ideology. To Islamists, Jihad is very much a physical struggle, spreading Islamification through the use or threat of terror, propaganda etc.  

Furthermore, establishing a 'caliphate' is an important part of the Islamist ideology. The idea of a caliphate means the deterritorialization of the Middle East that was set up by the Western countries after World War I, creating an Islamic State, which is the main priority of terrorist organizations such as ISIL (Islamic State of Iraq and the Levant). This demand for a pure and united Arab nation is also derived from the Quran, however, by Islamist extremists, this is interpreted as enforcing the Islamic faith around the world. For example, the striving for religious purity is documented as in 2014, there was mass persecution of Christians, Yazidis and Shia Muslims in Northern Iraq, and more recently in February 2015 as 21 Coptic Christians were executed by the terrorist group. This form of purity is based on the eradication of those who do not share the same beliefs, and in this sense, the demand for purity could be seen as an extreme form of identity politics, whereby the Middle East has been oppressed by the West (mainly the United States) for a long time, showing classic Imperialistic signs, that terrorist groups are considered in these nations as 'freedom fighters' in many people's minds. 

Finally, it appears that terrorist agenda is to reach a political end by definition, but it's clear that the search for purity is possibly a means to an end for Islamist Terrorism, as while Islam takes it's teachings from the Quran, including the idea of 'Shariah Law' which dictates a cruel legal and punishment system, it is widely debated that the core teachings of the Quran that are open to interpretation is seen as primarily peaceful, so it seems as if the terrorist search for purity is not only convoluted but a tactic for gaining power and damaging conflicting ideologies or powers.