Note, first, that we generally speak of a perspective on a subject, not of it.
There is no graceful way to deploy one noun phrase as object of two different prepositions heading complements of two different nouns, adjectives or verbs. There is a notoriously graceless way, which is the construction you suggest:
I will widen my perspective on and approach to complex interdisciplinary research.
This ugly construction infuriates writers and troubles linguists, because it conjoins non-constituent collocations. Nonetheless, it is universally accepted in academic and bureaucratic/legal dialects, so you need not hesitate to employ it.
The alternative is to rewrite. If you cannot find an alternative to one of your nouns which takes the same complement as the other, you may conjoin at a higher level:
I will widen my perspective on complex interdisciplinary research and my approach to it.
A POSSIBLY IMPERTINENT OBSERVATION:
If I might take a step back from the immediate question: My own experience is that awkward constructions like this usually betray some defect in my underlying metaphors. You should think about, for instance, what exactly you mean by widening your perspective and widening your approach. Is it possible that widen is just an approximation to what you really mean? —which might be something as remote from your first-order expression as this:
I will take a wider view of complex interdisciplinary research in order to find more productive approaches to the problem at hand.