A number of tense-forms are possible after 'hope' for present or future time. 'Would' is not normally used, except in such constructions as 'would like', 'would rather'.
I hope he is enjoying his new job. Present time.
I hope he likes his new job. Present or future time, depending on context.
I hope he will like his new job. Future time.
I hope he is coming/is going to come tomorrow. Future time.
Other tenses are possible with other time references:
I hope he has passed his test.
I hope he got home safely last night.
Student or Learner