ARMCHAIR SCHOLAR WARNING: Though I've become an enthusiast of various kinds of mystical thought, know that at least for Islamic spirituality I am far from an expert. What I present here I gleaned largely from books by the orientalist and scholar of comparative religion Henry Corbin. So if you find mistakes here, don't be surprised--I'm still kind of new to this topic.
Mormonism is, without a doubt, an odd religion. Not only do we believe in seer stones and Liahonas, but we also affirm the existence of a human God, complete with hair, a nose, and toenails. But by far the oddest belief we hold is that of a hierarchy of gods, or rather, that just as God was once a man, you can someday become a God or a Goddess yourself.
However, at least with the last belief, it may console the self-conscious believer to know that another, completely independent belief system believes in almost exactly the same thing. This system is the Ismāʿīlī sect of Shia Islam, and though it may seem exotic, there are actually few things closer to the Mormon worldview.
As opposed to many other branches of Islam, the Ismāʿīlīs place their emphasis on esoteric and mystically-minded interpretations of the law and the Qur'an. As such, they have a concept of ever-deepening spiritual interpretation (what they call ta'wil) that they hold in especially high regard. But ta'wil doesn't just mean the act of interpretation. At least in an Ismāʿīlī context, ta'wil refers to the act of bringing something back to its eternal origin. To do this with a sacred text means seeing through the text to the aspects of divinity that manifest through it, but it is far from a mere textual exercise--even events in the world can be spiritually interpreted this way.
Here we have the first inking of a connection with Mormonism: in both perspectives all things strive to go "upwards" toward God, to connect in actuality with the eternal potential they had pre-existently ("the measure of its creation," to quote D&C 88:19). In the Ismāʿīlī perspective, this happens because what they call "the angel of humanity" was himself cut off from the ultimate divine source and thus longed "nostalgically" to go back to it. Much like Joseph Smith's presentation of God the Father, this angel of humanity acts as the creator of this world, its window to divinity, and the "image" off of which each person's own angelic potential was based. Again in alignment with Mormonism, this angel is not ultimate--he is actually relatively far down the chain of spiritual hierarchy that pinnacles in who Muslims call Allah.
And though I don't know much about this particular aspect of Ismāʿīlī belief, I have read that once a person dies, they believe he or she is able to indefinitely progress with that angel of humanity back to the divine source from which all emanates. Hence here we have something strikingly similar to the Mormon doctrine of eternal progression.
It's worth noting, however, that this indefinite spiritual progression happens in a way you might not expect. The process actually occurs through ta'wil, or the attempt to return to the source of something's being by ever-deepening interpretation. Ta'wil lets one encounter the divinity manifest through an object, animal, or person; instead of being opaque, they become transparent to divinity. By doing this with your life, you get "drawn up" to divinity, while divinity becomes glorified in you. Thus ta'wil, as a method of seeing the source of things through an object of perception, lets one reconnect with that source both here and in eternity.
Though people might find it distasteful to believe that the process of eternal progression happens through ever-deepening interpretation (i.e. in a Mormon context, too), I don't see any reason why somebody shouldn't. After all, Joseph Smith talks about "all things [having] their likeness, that they may accord one with another--that which is earthly conforming to that which is heavenly," and insists that rituals like baptism for the dead act as such a salvific window to heaven. In fact, to view the concept of eternal progression in terms of ever deepening insight into the nature of reality echoes Joseph Smith's oft-repeated declaration that man is saved through knowledge.
In fact, if one recognizes the eerie similarity that the Ismāʿīlī system has to Mormon doctrine (it strikes me as hard not to), one might gain insight by reading certain Mormon concepts in an Ismāʿīlī way. For instance, to the question of how "progression" can occur in a timeless state, we might turn to the Ismāʿīlī notion that "past," "present," and "future" mean different things when separated from lower, "earthly" time. Namely, in higher spiritual echelons the present simply means eternal being. Similarly, in such a state the future refers to that eternal being in an active sense, while the past refers to the same eternal being as utilized toward action. Naturally this is quite difficult to understand, but the gist of it is that eternal time is nowhere near as absurd as time is here (in which we chase forever after ever-receding future satisfaction); there the past, present, and future exist together, only differentiated in mode of being.
One might also compare the Mormon notion of "many worlds" to its Ismāʿīlī parallel. These Muslims also believe in many worlds that are all presided over by the same overarching divinity, but they avoid the common Mormon way of interpreting these worlds as literal planets (which I think is in error). Instead, they declare that they are independent emanations from other angels, which themselves are emanations from higher ones. In other words, these worlds aren't just separated by quantitative distance--they are different from our mode of being in a qualitative way (in which one could never travel there in a spaceship, for instance). To use a crude metaphor, one could compare them to the parallel universes of science fiction, though that image is too materialist for my taste. But most interestingly, they declare that these parallel worlds (which, by the way, can be "higher" or "lower" in the spiritual hierarchy) connect with ours through the spiritual archetypes/aspects of divinity that manifest in both. It's the same image in both places, so to speak.
Moreover, one might productively understand Mormon conceptions of gender difference as do the Ismāʿīlīs. For them, the feminine principle represented the esoteric side of reality, the principle that, though hidden, is so because it is closer to God. Naturally, then, the masculine principle would represent the exoteric principle, that which is concerned with outer appearance and literalisms (Gaston Bachelard, whom I quoted in a recent post, says something similar in his Poetics of Reverie). As I mentioned in my post Letters to a Doubter: On Gender and Sexuality, one can read human anatomy in this way--though the woman's reproductive organs are more hidden from sight than the man's, they represent the origin of all human life. One might even say that, just as human life has an invisible origin in the woman, the human spirit itself rests upon and derives its entire being from the less-talked-about feminine aspect of the divine. Such would be an understanding of the feminine that, while symbolically manifest in the female body, does not necessarily determine the woman's personality or destiny (again, see the post mentioned above).
Finally, one might compare the Ismāʿīlī idea of the "The Final Imam," who will come at to initiate the final resurrection, with the Mormon/generic Christian idea of the Christ of the Second Coming. For Ismāʿīlīs, the final Imam will come as the embodied summation of the entire human spiritual community. In that sense, he is the manifestation in the world of the "angel of humanity" mentioned above, of whom we are all images. Naturally this conures images of "the body of Christ" and of Paul saying that "when Christ, who is our life, shall appear, then shall ye also appear with him in glory." (Colossians 3:4). In a very real sense, we are in Christ. Though this might strike you as needlessly sectarian, even the Book of Mormon says that "in Christ there should come every good thing," implying that Christ is indeed the recapitulation of all the good in the world. What comes from this observation, one which says that Christ's coming is really the coming to earth of the fullness of all good, is that we contribute toward that coming's fulfillment by manifesting that good within ourselves. Indeed, you could say that every manifestation of that good brings Christ back in miniature. The Ismāʿīlīs certainly thought this of their Imam, of whom they declared "may we be those who bring about the transfiguration of the world." But lest you think I'm eschewing or the personality of Christ, know that, as far as I'm aware, the Ismāʿīlīs thought of their Imam as a very real, concrete figure to come (whose body they spoke of in very real ways). Likewise, though I think that Christ will bring all good things with Him when he comes, I don't think that's opposed to the idea of his literal body. Oppositions of that sort are only exclusive to the mind attuned solely to the everyday.
In summary, l think we as Mormons have a lot to learn from the esoteric parts of Islam. As If have tried to show above, I believe that these two systems represent different perspectives of the same eternal landscape, if you will. By comparing the two, we may get a better intuitive grasp of what the landscape looks like in itself.