The question you asked is more of a philosophical one instead of a mathematical one. Your question depends on the more general and fascinating question, “What is an algorithm?” Let me explain:
Everything involving computer science and computation in general can be thought of in terms of algorithms: recursive functions, programming languages, programs, computers, operating systems, etc. In the early days, the notion ‘all good algorithms can be translated into some halting Turing machine’ would have been widely accepted. Today, however, halting Turing machines do not maintain the high status they once held. There is an apparent information gap when we try to reduce so called interactive and parallel algorithms into halting Turing machines. Intuitively this can be understood as trying to reduce a windows operating system into a halting Turing machine (sounds ridiculous right?). The ridiculousness of this task stems from the fact that, ideally, an operating system like windows should never halt. You don’t want to be on Microsoft word and suddenly the algorithm ‘halts’ without being given specific instructions. Operating systems are definitely algorithms (a system of specific instructions for given inputs), but for long drawn out projects we might want our operating system to run indefinitely. So in regards to your question “Must an algorithm terminate?” in my personal opinion the answer is no. Similarly, in regards to the question, “Are all algorithms translatable into some halting Turing machine?” my answer is also no.
Many of the ideas I have suggested here were brought to my attention by the work of algorithm master Yuri Gurevich (Russian mathematician and computer scientist who now works for Microsoft). Gurevich has attempted to generalize the concept of algorithm with something he calls ‘abstract state machines’: a system for formalizing algorithms including the non-Turing-machine-convertible algorithms (like interactive and parallel algorithms). ‘Abstract state machines’ have been met with wide acclaim and there are learned societies focused solely on them.
To find more information on the work of Yuri Gurevich check out these two lectures:
Yuri Gurevich, Church Turing Thesis: Story & Recent Progress (2009)
Yuri Gurevich, What’s an Algorithm? (2011)