Commitments and Contingencies - Additional Information (Details) $ in Millions	3 Months Ended
Mar. 31, 2022 USD ($) Leases
Lessee, Lease, Description [Line Items]
Number of operating lease \| Leases	1
Operating lease initial term	4 years 5 months
Discount rate	6.25%
Operating lease remaining lease term	3 years 2 months
ABL Bio Agreements Member
Lessee, Lease, Description [Line Items]
Milestone payments \| $	$ 6.0

Milestone payments.

No definition available.

Name:	cmpx_MilestonePayments
Namespace Prefix:	cmpx_
Data Type:	xbrli:monetaryItemType
Balance Type:	credit
Period Type:	duration

- Definition

Number of operating lease.

+ References

No definition available.

+ Details

Name:	cmpx_NumberOfOperatingLease
Namespace Prefix:	cmpx_
Data Type:	xbrli:integerItemType
Balance Type:	na
Period Type:	duration

- Definition

Line items represent financial concepts included in a table. These concepts are used to disclose reportable information associated with domain members defined in one or many axes to the table.

+ References

No definition available.

+ Details

Name:	us-gaap_LesseeLeaseDescriptionLineItems
Namespace Prefix:	us-gaap_
Data Type:	xbrli:stringItemType
Balance Type:	na
Period Type:	duration

- Definition

Discount rate used by lessee to determine present value of operating lease payments.

+ References

Reference 1: http://www.xbrl.org/2003/role/exampleRef
-Publisher FASB
-Name Accounting Standards Codification
-Topic 842
-SubTopic 20
-Section 50
-Paragraph 3
-Subparagraph (c)(3)
-URI http://asc.fasb.org/extlink&oid=123408670&loc=SL77918673-209980

+ Details

Name:	us-gaap_LesseeOperatingLeaseDiscountRate
Namespace Prefix:	us-gaap_
Data Type:	dtr-types:percentItemType
Balance Type:	na
Period Type:	instant

- Definition

Remaining lease term of operating lease, in 'PnYnMnDTnHnMnS' format, for example, 'P1Y5M13D' represents reported fact of one year, five months, and thirteen days.

+ References

Reference 1: http://www.xbrl.org/2009/role/commonPracticeRef
-Publisher FASB
-Name Accounting Standards Codification
-Topic 842
-SubTopic 20
-Section 50
-Paragraph 3
-Subparagraph (a)(1)
-URI http://asc.fasb.org/extlink&oid=123408670&loc=SL77918673-209980

+ Details

Name:	us-gaap_LesseeOperatingLeaseRemainingLeaseTerm
Namespace Prefix:	us-gaap_
Data Type:	xbrli:durationItemType
Balance Type:	na
Period Type:	instant

- Definition

Term of lessee's operating lease, in 'PnYnMnDTnHnMnS' format, for example, 'P1Y5M13D' represents reported fact of one year, five months, and thirteen days.

+ References

Reference 1: http://www.xbrl.org/2009/role/commonPracticeRef
-Publisher FASB
-Name Accounting Standards Codification
-Topic 842
-SubTopic 20
-Section 50
-Paragraph 3
-Subparagraph (a)(3)
-URI http://asc.fasb.org/extlink&oid=123408670&loc=SL77918673-209980

+ Details

Name:	us-gaap_LesseeOperatingLeaseTermOfContract
Namespace Prefix:	us-gaap_
Data Type:	xbrli:durationItemType
Balance Type:	na
Period Type:	instant

- Details

Name:	us-gaap_TypeOfArrangementAxis=cmpx_AblBioAgreementMember
Namespace Prefix:
Data Type:	na
Balance Type:
Period Type: